#7566070

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	#45134	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2040 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,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, 7473266);