#6868396

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1255 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)(25,26,29,34,30,35,41,32)(27,31,38,42,36,44,50,45)(28,33,40,48,39,47,52,46)(37,43,51,53,49,54,56,55);(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)(26,31)(28,37)(29,38)(30,36)(32,45)(33,43)(34,42)(35,44)(39,49)(40,51)(41,50)(46,55)(47,54)(48,53)(52,56);(2,6)(7,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(26,32)(27,36)(29,41)(31,42)(33,46)(34,35)(37,49)(40,52)(43,53)(44,45)(47,48)(54,55);(1,3,5,4)(2,7,6,8)(10,16)(14,17)(15,22)(21,23)(25,28)(26,33)(27,37)(29,40)(30,39)(31,43)(32,46)(34,48)(35,47)(36,49)(38,51)(41,52)(42,53)(44,54)(45,55)(50,56);(1,4,5,3)(2,7,6,8)(11,19)(14,17)(18,24)(21,23)(27,36)(28,39)(31,44)(33,47)(38,50)(40,52)(42,45)(46,48)

Polycyclic presentation:

2^9|a^2=g,c^2=f,d^2=hi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,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, 6775592);