#72337293

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁶
commutator	C₂³	C₂⁵×C₄
prime radical	C₂⁴×H(5,𝔽₂)	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

248125 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=g,c^2=gh,e^2=gij,b^a=bj,c^b=ch,d^b=dhi,d^c=dhij,e^b=eij,e^c=ehij,e^d=ehi,f^b=fj,f^c=fj,f^e=fj

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72337295
maximal normal subgroup of	#72337295
maximal quotient of	#72337319, #72337392

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, 2, F.7);;
gap> SetPower(coll, 3, F.7*F.8);;
gap> SetPower(coll, 5, F.7*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);