#433106

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

1304 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340302);