#463628

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
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

828 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 370824);