#477530

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
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	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1046 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384726);