#7613560

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1958 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7520756);