#472504

order	512
nilpotency class	3
isoclinism	isoclinic class of #455110
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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

902 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egh,c^2=ef,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^b=di,d^c=de,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, 379700);