#6614951

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	#37893	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1555 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fgh,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6522147);