#7565212

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1795 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)(7,11)(8,12)(9,10)(13,14)(15,21)(16,28)(17,34)(18,24)(19,36)(20,25)(22,32)(23,27)(26,39)(29,37)(30,41)(31,40)(33,43)(35,38)(42,44);(1,2)(3,4)(6,10)(11,12)(13,15,20,27)(14,21,25,23)(16,29,33,30)(17,19,35,18)(22,39,40,44)(24,34,36,38)(26,31,42,32)(28,37,43,41);(5,7,9,8)(6,11,10,12)(13,16,19,32,20,33,18,31)(14,22,24,28,25,40,36,43)(15,26,17,29,27,42,35,30)(21,37,38,39,23,41,34,44);(2,4)(5,7,9,8)(6,11,10,12)(13,17,19,27,20,35,18,15)(14,23,24,34,25,21,36,38)(16,30,32,26,33,29,31,42)(22,37,28,39,40,41,43,44);(5,8,9,7)(6,11,10,12)(13,18,20,19)(14,24,25,36)(15,17,27,35)(16,31,33,32)(21,34,23,38)(22,28,40,43)(26,29,42,30)(37,44,41,39)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,d^2=fg,e^2=fi,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^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, 7472408);