#435083

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

873 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=f,c^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=dei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=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, 342279);