#7614493

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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1800 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)(13,14,20,24)(15,21,28,37)(16,29,31,40)(17,18,23,19)(22,33,39,32)(25,38,41,43)(26,34,27,36)(30,35,42,44);(2,4)(6,10)(11,12)(13,15,20,28)(14,21,24,37)(16,30,31,42)(17,34,23,36)(18,27,19,26)(22,38,39,43)(25,33,41,32)(29,35,40,44);(5,7,9,8)(6,11,10,12)(13,16,18,32,20,31,19,33)(14,22,17,29,24,39,23,40)(15,25,26,30,28,41,27,42)(21,35,36,38,37,44,34,43);(1,2,3,4)(13,17,18,24,20,23,19,14)(15,21,26,36,28,37,27,34)(16,29,32,39,31,40,33,22)(25,35,30,38,41,44,42,43);(5,8,9,7)(6,11,10,12)(16,31)(22,39)(25,41)(29,40)(30,42)(32,33)(35,44)(38,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=hi,e^2=f,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, 7521689);