#438833

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

749 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(2,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,21,26)(19,29,32,23)(20,34,36,24)(22,27,39,28)(25,37,45,38)(30,40,46,44)(31,47,35,43)(33,42,48,41);(1,2)(3,7)(4,8)(5,6)(17,19,22,33,21,32,39,48)(18,23,27,41,26,29,28,42)(20,35,37,46,36,31,38,30)(24,43,25,44,34,47,45,40);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20)(18,24)(19,30)(21,36)(22,38)(23,40)(25,28)(26,34)(27,45)(29,44)(31,33)(32,46)(35,48)(37,39)(41,47)(42,43);(1,4,5,3)(2,7,6,8)(17,20)(18,25)(19,31)(21,36)(22,37)(23,40)(24,28)(26,45)(27,34)(29,44)(30,33)(32,35)(38,39)(41,43)(42,47)(46,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=ei,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, 346029);