#6628597

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	C₂×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1501 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)(3,8)(4,6)(5,7)(9,10,13,14)(11,15,12,16)(17,18)(19,24)(20,29)(21,25)(22,32)(23,26)(27,31)(28,30);(1,3,5,4)(2,6,7,8)(10,14)(11,12)(17,19,23,22)(18,24,26,32)(20,25,30,27)(21,28,31,29);(9,11,13,12)(10,15,14,16)(17,20,22,25,23,30,19,27)(18,21,24,29,26,31,32,28);(1,4,5,3)(2,6,7,8)(11,12)(15,16)(17,21,23,31)(18,25,26,27)(19,28,22,29)(20,24,30,32);(9,12,13,11)(10,16,14,15)(17,19,23,22)(18,24,26,32)(20,27,30,25)(21,29,31,28)

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,c^2=ghi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,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, 6535793);