#7614477

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	C₂×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1847 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)(15,25)(16,30)(17,18)(20,26)(21,37)(22,23)(27,41)(28,29)(31,38)(32,33)(34,44)(35,36);(2,4)(6,10)(11,12)(13,14,19,24)(15,26,28,36)(16,31,33,37)(17,23,18,22)(20,29,35,25)(21,30,38,32)(27,34,41,44)(39,43,42,40);(5,7,9,8)(6,11,10,12)(13,15,17,25,19,28,18,29)(14,20,22,26,24,35,23,36)(16,27,30,39,33,41,32,42)(21,34,37,40,38,44,31,43);(1,2,3,4)(13,16,17,30,19,33,18,32)(14,21,22,37,24,38,23,31)(15,27,25,39,28,41,29,42)(20,34,26,40,35,44,36,43);(5,8,9,7)(6,11,10,12)(15,28)(20,35)(25,29)(26,36)(27,41)(34,44)(39,42)(40,43)

Polycyclic presentation:

2^9|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, 7521673);