#6579559

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1313 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486755);