#6671833

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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2319 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)(15,22)(16,25)(17,23)(18,20)(19,27)(21,24)(26,28);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(14,20)(21,28)(22,25)(23,27);(2,4)(13,15,18,16)(14,21,20,28)(17,24,19,26)(22,23,25,27);(5,7,8,11)(6,10,9,12)(13,16,18,15)(14,22,20,25)(17,24,19,26)(21,23,28,27);(2,4)(6,9)(10,12)(13,17,18,19)(14,23,20,27)(15,24,16,26)(21,25,28,22)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6579029);