#476603

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×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1033 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^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=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, 383799);