#6671960

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2298 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6579156);