#439375

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

897 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)(6,7)(9,10,13,15)(11,14,12,16)(17,18)(19,24)(20,22)(21,29)(23,25)(26,27)(28,30)(31,32);(1,2,4,5)(3,7,8,6)(9,11,13,12)(10,14,15,16)(17,19,27,32,22,28,23,29)(18,24,26,31,20,30,25,21);(9,12,13,11)(10,14,15,16)(17,20,22,18)(19,21,28,31)(23,25,27,26)(24,32,30,29);(1,3,4,8)(2,6,5,7)(17,21)(18,19)(20,28)(22,31)(23,30)(24,27)(25,29)(26,32)

Polycyclic presentation:

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

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, 346571);