#6673287

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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

2144 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:

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

Polycyclic presentation:

2^9|a^2=i,b^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,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, 6580483);