#6804830

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1207 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fgi,d^2=f,e^2=hi,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^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, 6712026);