#6843732

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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	#42855	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1454 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:

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

Polycyclic presentation:

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

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, 6750928);