#471526

order	512
nilpotency class	3
isoclinism	isoclinic class of #455110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

1051 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:

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

Polycyclic presentation:

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

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, 378722);