#6489509

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470374
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1364 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:

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

Polycyclic presentation:

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

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, 6396705);