#471728

order	512
nilpotency class	3
isoclinism	isoclinic class of #455107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

815 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=egh,c^2=e,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,f^c=fi,g^c=gi

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, 378924);