#6742014

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,7)(3,8,4,6)(9,10,12,14)(11,15,16,13)(18,23)(19,20)(26,31)(28,29);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,18,21,23)(19,27,20,25)(22,31,30,26)(24,29,32,28);(10,14)(13,15)(17,19)(18,24)(20,21)(22,29)(23,32)(25,26)(27,31)(28,30);(1,3,5,4)(2,6,7,8)(10,14)(13,15)(17,20,21,19)(18,25,23,27)(22,29,30,28)(24,31,32,26);(1,4,5,3)(2,6,7,8)(11,16)(13,15)(18,26)(22,30)(23,31)(24,25)(27,32)(28,29)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=fgi,e^2=fg,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, 6649210);