#7062813

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,f^b=fi,d^c=dhi,e^c=ei,f^d=fi

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, 6970009);