#6830884

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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1594 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=fi,e^2=ghi,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, 6738080);