#6549005

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533087
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	#2011	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1561 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6456201);