#6629757

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629016
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2214 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,d^b=di,e^b=ei,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536953);