#431414

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
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

773 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)(10,13)(11,15)(17,18)(19,28)(20,24)(21,26)(22,27)(23,32)(25,31)(29,30);(1,3,5,8)(2,6,4,7)(9,10,12,13)(11,14,15,16)(17,18,21,26)(19,25,20,30)(22,32,23,27)(24,31,28,29);(9,11,12,15)(10,14,13,16)(17,19)(18,24)(20,21)(22,31)(23,29)(25,32)(26,28)(27,30);(1,4,5,2)(3,6,8,7)(10,13)(14,16)(17,20)(18,25)(19,21)(22,31)(23,29)(24,32)(26,30)(27,28)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=fi,d^2=ef,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^b=gi,g^c=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 338610);