#7613604

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	#45412	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1966 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)(13,14,19,23)(15,20,26,36)(16,27,25,40)(17,30,32,43)(18,34,28,22)(21,37,39,33)(24,38,41,44)(29,42,31,35);(2,4)(6,9)(10,12)(13,15,19,26)(14,20,23,36)(16,28,25,18)(17,31,32,29)(21,38,39,44)(22,40,34,27)(24,37,41,33)(30,35,43,42);(5,7,8,11)(6,10,9,12)(13,16,28,26,19,25,18,15)(14,20,34,40,23,36,22,27)(17,29,37,41,32,31,33,24)(21,35,43,44,39,42,30,38);(1,2,3,4)(13,17,28,37,19,32,18,33)(14,21,34,43,23,39,22,30)(15,24,16,29,26,41,25,31)(20,35,40,44,36,42,27,38);(6,9)(10,12)(17,32)(21,39)(24,41)(29,31)(30,43)(33,37)(35,42)(38,44)

Polycyclic presentation:

2^9|a^2=i,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=cgi,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7520800);