#6477438

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470067
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2965 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6384634);