#6691075

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667508
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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1490 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)(3,6)(4,8)(5,7)(9,10)(11,13)(12,15)(14,16)(19,28)(21,30)(24,32)(26,29);(1,3,5,4)(2,6,7,8)(9,11,13,10)(12,15,16,14)(17,18)(19,29)(20,25)(21,24)(22,27)(23,31)(26,28)(30,32);(1,4,5,3)(2,6,7,8)(10,11)(14,15)(17,19)(18,24)(20,21)(22,28)(23,30)(25,26)(27,32)(29,31);(9,12,13,16)(10,14,11,15)(17,20,22,23)(18,25,27,31)(19,21,28,30)(24,26,32,29);(17,21,22,30)(18,26,27,29)(19,20,28,23)(24,25,32,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6598271);