#6676384

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

1967 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=i,d^2=f,e^2=f,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, 6583580);