#6573899

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2645 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,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, 6481095);