#6843780

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	#42619	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1501 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=fhi,d^2=gi,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, 6750976);