#6580171

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1593 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,d^2=fi,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, 6487367);