#6629306

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628851
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4092 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,g^c=gi,e^d=ei

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, 6536502);