#6551924

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 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)(9,10,15,14)(11,12,13,16)(17,18,21,25,20,24,29,23)(19,26,28,31,27,22,30,32);(1,3,5,4)(2,6,7,8)(9,11,15,13)(10,16,14,12)(17,19,20,27)(18,22,24,26)(21,28,29,30)(23,31,25,32);(1,4,5,3)(2,6,7,8)(9,12,15,16)(10,11,14,13)(18,23)(21,29)(22,31)(24,25)(26,32)(28,30);(9,13,15,11)(10,16,14,12)(18,24)(19,27)(23,25)(28,30);(9,14,15,10)(11,12,13,16)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6459120);