#7036370

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1257 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(2,6)(7,8)(9,10,13,16)(11,14,18,22)(12,15,20,24)(17,21,19,23)(25,26)(27,37)(28,34)(29,35)(30,36)(31,46)(32,40)(33,42)(38,54)(39,50)(41,48)(43,52)(44,49)(45,53)(47,55)(51,56);(1,2,5,6)(3,7,4,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27)(26,32)(28,41)(29,44)(30,40)(31,39)(33,51)(34,54)(35,50)(36,37)(38,43)(42,55)(45,56)(46,49)(47,53)(48,52);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28)(26,33)(27,38)(29,42)(30,43)(31,45)(32,47)(34,46)(35,52)(36,53)(37,51)(39,55)(40,41)(44,56)(48,49)(50,54);(1,4,5,3)(2,7,6,8)(12,20)(15,24)(17,19)(21,23)(25,28)(26,34)(27,38)(29,45)(30,43)(31,42)(32,48)(33,46)(35,53)(36,52)(37,54)(39,56)(40,41)(44,55)(47,49)(50,51);(10,16)(11,18)(15,24)(17,19)(25,29,30,31)(26,35,36,46)(27,39,40,44)(28,42,43,45)(32,49,37,50)(33,52,53,34)(38,55,41,56)(47,48,51,54)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=fi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6943566);