#6765112

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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)(25,26,30,34)(27,35,38,51)(28,32,40,47)(29,43,36,33)(31,44,46,39)(37,50,41,54)(42,56,52,49)(45,55,48,53);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(28,40)(32,47)(37,41)(42,52)(45,48)(49,56)(50,54)(53,55);(2,6)(3,7)(25,27,36,44,30,38,29,39)(26,31,43,51,34,46,33,35)(28,41,52,53,40,37,42,55)(32,48,56,50,47,45,49,54);(10,15)(14,17)(16,23)(22,24)(25,28)(26,32)(27,37)(29,42)(30,40)(31,45)(33,49)(34,47)(35,50)(36,52)(38,41)(39,53)(43,56)(44,55)(46,48)(51,54);(3,7)(5,8)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(27,38)(28,40)(31,46)(32,47)(35,51)(39,44)(42,52)(49,56)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=ghi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,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, 6672308);