#7613683

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#45136	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1978 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)(16,28)(17,25)(19,24)(20,29)(21,31)(26,40)(27,32)(30,35)(33,41)(34,39)(36,42);(2,4)(6,9)(10,12)(13,14,18,22)(15,24,27,39)(16,29,30,42)(17,31,25,21)(19,32,34,23)(20,35,36,28)(26,41,40,33)(37,38,43,44);(5,7,8,11)(6,10,9,12)(13,15,25,32,18,27,17,23)(14,19,31,39,22,34,21,24)(16,26,35,43,30,40,28,37)(20,33,42,44,36,41,29,38);(1,2,3,4)(13,16,25,35,18,30,17,28)(14,20,31,42,22,36,21,29)(15,26,32,43,27,40,23,37)(19,33,39,44,34,41,24,38);(6,9)(10,12)(15,27)(19,34)(23,32)(24,39)(26,40)(33,41)(37,43)(38,44)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,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,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, 7520879);