#6629158

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 ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4202 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)(13,14,18,23)(15,19,26,33)(16,21,27,35)(17,20,29,22)(24,31,37,42)(25,30,39,32)(28,34,40,36)(38,41,44,43);(6,9)(10,12)(13,15,18,26)(14,19,23,33)(16,24,27,37)(17,25,29,39)(20,30,22,32)(21,31,35,42)(28,38,40,44)(34,41,36,43);(5,7,8,11)(6,10,9,12)(14,20)(17,29)(19,30)(21,34)(22,23)(25,39)(28,40)(31,41)(32,33)(35,36)(38,44)(42,43);(2,4)(13,16)(14,21)(15,24)(17,28)(18,27)(19,31)(20,34)(22,36)(23,35)(25,38)(26,37)(29,40)(30,41)(32,43)(33,42)(39,44);(16,27)(21,35)(24,37)(28,40)(31,42)(34,36)(38,44)(41,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,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, 6536354);