#6820804

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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1377 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)(3,6,4,8)(9,10,12,14)(11,15,16,13)(17,18,22,23)(19,21,26,30)(20,27,29,24)(25,32,31,28);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,21)(20,28)(22,26)(23,30)(24,31)(25,27)(29,32);(1,4,5,3)(2,6,7,8)(10,14)(11,16)(17,20,18,24,22,29,23,27)(19,25,30,32,26,31,21,28);(10,14)(13,15)(20,29)(24,27)(25,31)(28,32);(17,21,18,19,22,30,23,26)(20,28,24,25,29,32,27,31)

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,c^2=gh,e^2=fhi,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, 6728000);