#6686748

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1989 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6593944);