#6579683

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#36009	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,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, 6486879);