#7614428

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	#24129	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1837 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)(13,14,18,22)(15,23,26,40)(16,27,31,43)(17,32,30,21)(19,33,35,24)(20,36,38,28)(25,37,29,34)(39,44,42,41);(2,4)(6,10)(11,12)(15,24)(16,28)(17,30)(19,23)(20,27)(21,32)(25,29)(26,33)(31,36)(34,37)(35,40)(38,43);(5,7,9,8)(6,11,10,12)(13,15,17,24,18,26,30,33)(14,19,21,23,22,35,32,40)(16,29,28,44,31,25,36,41)(20,37,27,42,38,34,43,39);(1,2,3,4)(13,16,30,36,18,31,17,28)(14,20,32,43,22,38,21,27)(15,25,33,44,26,29,24,41)(19,34,40,42,35,37,23,39);(5,8,9,7)(6,11,10,12)(16,31)(20,38)(25,29)(27,43)(28,36)(34,37)(39,42)(41,44)

Polycyclic presentation:

2^9|a^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,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, 7521624);