#6764989

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1350 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6672185);