#6836916

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	#42438	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1704 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gh,d^2=fi,e^2=ghi,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, 6744112);