#6844044

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1340 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18,22,26)(19,23,25,21)(20,24,29,31)(27,30,32,28);(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,28)(21,26)(22,25)(24,27)(29,30)(31,32);(2,6)(7,8)(11,16)(13,15)(17,20,18,24,22,29,26,31)(19,27,21,28,25,32,23,30);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(20,29)(24,31)(27,32)(28,30);(1,4,5,3)(2,8,6,7)(17,21,26,19,22,23,18,25)(20,28,31,27,29,30,24,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6751240);