#7142921

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
prime radical	#45332	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1989 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7050117);