#450172

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
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

872 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)(11,13)(12,14)(15,16)(17,18,23,28,22,27,38,46)(19,29,32,26,21,24,31,40)(20,25,37,45,35,43,36,44)(30,47,33,42,34,39,48,41);(1,2)(3,7)(4,8)(5,6)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,33)(21,22)(23,32)(25,41)(26,46)(27,29)(28,40)(30,36)(31,38)(34,37)(35,48)(39,44)(42,43)(45,47);(1,3,5,4)(2,7,6,8)(10,14)(13,16)(17,20)(18,25)(19,30)(21,34)(22,35)(23,37)(24,39)(26,42)(27,43)(28,45)(29,47)(31,48)(32,33)(36,38)(40,41)(44,46);(1,4,5,3)(2,7,6,8)(17,21,22,19)(18,26,27,40)(20,34,35,30)(23,32,38,31)(24,28,29,46)(25,42,43,41)(33,36,48,37)(39,45,47,44)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=ei,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,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, 357368);