#6651538

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632048
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	#37334	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1573 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=h,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,e^b=ei,e^c=ei,g^c=gi

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, 6558734);