#6686756

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

1975 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)(13,14,18,19)(15,20,24,28)(16,25,27,21)(17,22,23,26);(5,6)(7,10)(8,12)(9,11)(13,15,18,24)(14,20,19,28)(16,26,27,22)(17,21,23,25);(1,2,3,4)(13,16,18,27)(14,21,19,25)(15,23,24,17)(20,22,28,26);(5,7,9,8)(6,10,11,12)(13,14,18,19)(15,20,24,28)(16,25,27,21)(17,22,23,26);(5,8,9,7)(6,10,11,12)(13,17)(14,22)(15,16)(18,23)(19,26)(20,25)(21,28)(24,27)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=g,d^2=fi,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, 6593952);