#460770

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1306 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=eg,b^2=f,c^2=i,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,g^c=gi

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, 367966);