#7613539

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

2000 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)(15,23)(16,27)(17,30)(19,24)(20,28)(21,32)(25,29)(26,33)(31,36)(34,37)(35,41)(38,44);(2,4)(6,9)(10,12)(13,14,18,22)(15,24,26,41)(16,28,31,44)(17,32,30,21)(19,33,35,23)(20,36,38,27)(25,37,29,34)(39,40,42,43);(5,7,8,11)(6,10,9,12)(13,15,17,23,18,26,30,33)(14,19,21,24,22,35,32,41)(16,29,27,42,31,25,36,39)(20,37,28,43,38,34,44,40);(1,2,3,4)(13,16,30,36,18,31,17,27)(14,20,32,44,22,38,21,28)(15,25,33,42,26,29,23,39)(19,34,41,43,35,37,24,40);(6,9)(10,12)(15,26)(19,35)(23,33)(24,41)(25,29)(34,37)(39,42)(40,43)

Polycyclic presentation:

2^9|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=cg,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, 7520735);