#6673235

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2197 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:

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

Polycyclic presentation:

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

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, 6580431);