#7036368

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=fi,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, 6943564);