#406733

order	512
nilpotency class	3
isoclinism	isoclinic class of #393999
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1308 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=f,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,g^c=gi

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, 313929);