#72335930

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
prime radical	#647656	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

34055 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:

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

Polycyclic presentation:

2^10|a^2=i,b^2=ij,f^2=gij,b^a=bg,c^a=cij,c^b=cij,d^a=dgh,d^b=dgh,e^a=eghij,e^b=eghij,f^a=fg,f^c=fij,f^d=fhi,f^e=fghj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336001
maximal normal subgroup of	#72336001
maximal quotient of	#72336390

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