#6794639

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667676
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	#2537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1460 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6701835);