#6499811

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499804
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3974 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)(13,14)(15,24)(16,21)(17,22)(18,31)(19,23)(20,26)(25,39)(27,34)(28,36)(29,37)(30,38)(32,35)(33,42)(40,43)(41,44);(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,25)(17,26)(18,32)(19,27)(21,33)(22,34)(23,35)(24,31)(28,40)(29,41)(30,42)(36,39)(37,43)(38,44);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,29)(14,21,23,37)(15,25,27,41)(17,30,18,28)(20,33,35,43)(22,38,31,36)(24,39,34,44)(26,42,32,40);(7,12)(9,11)(13,17,19,18)(14,22,23,31)(15,26,27,32)(16,28,29,30)(20,34,35,24)(21,36,37,38)(25,40,41,42)(33,39,43,44);(6,10)(9,11)(16,29)(21,37)(25,41)(28,30)(33,43)(36,38)(39,44)(40,42)

Polycyclic presentation:

2^9|c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6407007);