#438756

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

796 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)(9,10,12,14)(11,15,16,13)(17,18)(19,20)(21,24)(22,26)(23,30)(25,32)(27,29)(28,31);(1,2,4,5)(3,7,8,6)(17,19,21,27,22,28,32,23)(18,20,24,29,26,31,25,30);(9,11,12,16)(10,13,14,15)(17,20)(18,19)(21,30)(22,31)(23,24)(25,27)(26,28)(29,32);(1,3,4,8)(2,6,5,7)(17,20)(18,23)(19,24)(21,29)(22,31)(25,28)(26,27)(30,32)

Polycyclic presentation:

2^9|a^2=f,b^2=eg,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, 345952);