#6577833

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

2064 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(14,19)(15,22)(20,25)(21,24);(6,10)(11,12)(13,14)(15,23)(16,25)(17,19)(18,24)(20,27)(21,28)(22,26);(2,4)(13,15,17,22)(14,16,19,27)(18,20,28,25)(21,26,24,23);(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,24)(17,27)(18,26)(19,25)(21,22)(23,28);(5,8,9,7)(6,12,10,11)(14,21)(16,26)(18,28)(19,24)(20,25)(23,27)

Polycyclic presentation:

2^9|c^2=i,d^2=fi,e^2=fi,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, 6485029);