#7561167

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	#45094	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1851 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)(15,21)(16,28)(17,31)(18,23)(19,35)(20,26)(22,34)(24,29)(25,41)(27,37)(30,36)(32,38)(33,44)(39,43)(40,42);(1,2)(3,4)(6,9)(10,12)(13,15,18,26)(14,20,23,21)(16,29,27,19)(17,32,33,30)(22,39,40,41)(24,28,35,37)(25,42,43,34)(31,36,44,38);(5,7,8,11)(6,10,9,12)(13,16,29,26,18,27,19,15)(14,21,35,37,23,20,24,28)(17,32,42,25,33,30,34,43)(22,36,44,41,40,38,31,39);(2,4)(13,17,19,34,18,33,29,42)(14,22,24,31,23,40,35,44)(15,25,27,32,26,43,16,30)(20,36,37,39,21,38,28,41);(6,9)(10,12)(14,23)(17,33)(20,21)(24,35)(25,43)(28,37)(30,32)(34,42)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=g,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^d=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, 7468363);