#6868609

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1239 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,33,30,35,29,36)(27,32,43,45,39,47,38,48)(28,34,44,49,41,52,42,51)(37,46,54,55,40,50,53,56);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23)(25,27,30,39)(26,32,35,47)(28,40,41,37)(29,38,31,43)(33,45,36,48)(34,50,52,46)(42,54,44,53)(49,56,51,55);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,33)(29,31)(32,45)(34,49)(35,36)(38,43)(42,44)(46,55)(47,48)(50,56)(51,52)(53,54);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(25,28)(26,34)(27,37)(29,42)(30,41)(31,44)(32,46)(33,49)(35,52)(36,51)(38,53)(39,40)(43,54)(45,55)(47,50)(48,56);(1,4,5,3)(2,7,6,8)(11,19)(14,17)(18,24)(21,23)(26,35)(28,41)(32,47)(33,36)(37,40)(42,44)(45,48)(53,54)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6775805);