#6630217

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628851
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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2156 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537413);