#6608966

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1462 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)(3,7)(4,8)(5,6)(9,10)(11,16)(12,14)(13,15)(17,18,22,19)(20,27,30,32)(21,24,28,25)(23,29,26,31);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,22,18)(20,23,30,26)(21,24,28,25)(27,29,32,31);(1,3,5,4)(2,7,6,8)(10,15)(11,12)(17,20,28,29,22,30,21,31)(18,23,24,32,19,26,25,27);(1,4,5,3)(2,8,6,7)(9,12,13,11)(10,14,15,16)(17,21,22,28)(18,24,19,25)(20,29,30,31)(23,27,26,32);(11,12)(14,16)(20,30)(23,26)(27,32)(29,31)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fgi,d^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei,e^d=ehi

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, 6516162);