#72232292

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁶
commutator	C₂³	C₂⁵×C₄
prime radical	C₂×#35222	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

31973 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

(5,6,8,10)(7,11,12,9)(15,19)(16,18)(22,25)(26,27);(5,7)(6,9)(8,12)(10,11)(13,14)(15,16)(17,20)(18,19)(22,25)(26,27);(6,10)(9,11)(13,15,17,19)(14,16,20,18)(22,25)(26,27);(6,10)(9,11)(15,19)(16,18)(21,22,24,25)(23,26,28,27);(5,7)(6,11)(8,12)(9,10)(13,15)(14,18)(16,20)(17,19)(21,23)(22,26)(24,28)(25,27);(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,16,17,18)(14,19,20,15)(21,22)(23,27)(24,25)(26,28)

Polycyclic presentation:

2^10|a^2=j,c^2=hj,d^2=hij,f^2=g,b^a=bj,c^a=ch,d^a=dhi,d^b=dhij,d^c=di,e^a=ehj,e^b=ehj,e^c=ehj,e^d=ehj,f^a=fij,f^b=fi,f^c=fij,f^d=fij,f^e=fij

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232371, #72236722, #72236750, …
maximal normal subgroup of	#72232371, #72236722, #72236750, …
maximal quotient of	C₂×#72232292

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.10);;
gap> SetPower(coll, 3, F.8*F.10);;
gap> SetPower(coll, 4, F.8*F.9*F.10);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);