#6622210

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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1429 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:

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

Polycyclic presentation:

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

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, 6529406);