#7566066

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	#45332	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1999 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=fg,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, 7473262);