#6573518

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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 ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4536 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6480714);