#409207

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

870 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)(3,6)(4,5)(7,8)(9,10,13,16)(11,15,12,14)(17,18)(19,26)(20,30)(21,24)(22,25)(23,27)(28,31)(29,32);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,18,21,30,22,25,20,24)(19,23,28,32,29,31,27,26);(9,12,13,11)(10,15,16,14)(17,18)(19,23)(20,24)(21,30)(22,25)(26,27)(28,32)(29,31);(1,3,4,7)(2,6,5,8)(17,19,20,27,22,29,21,28)(18,23,24,26,25,31,30,32)

Polycyclic presentation:

2^9|a^2=f,b^2=fhi,c^2=f,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,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, 316403);