#6629689

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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2289 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:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=i,d^2=f,e^2=i,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, 6536885);