#7105580

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1407 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=fh,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 7012776);