#472500

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	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

916 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)(19,23)(20,32)(21,27)(22,25)(24,34)(26,28)(29,41)(30,40)(31,38)(33,47)(35,42)(36,43)(37,45)(39,44)(46,48);(1,3,6,4)(2,7,5,8)(9,11,12,15)(10,13,14,16)(17,19,28,44,22,31,21,30)(18,23,26,39,25,38,27,40)(20,33,34,29,36,45,35,46)(24,41,43,37,42,48,32,47);(1,4,6,3)(2,7,5,8)(17,20,21,34,22,36,28,35)(18,24,26,32,25,42,27,43)(19,29,30,45,31,46,44,33)(23,37,39,41,38,47,40,48);(1,5,6,2)(3,8,4,7)(10,14)(13,16)(18,25)(23,38)(24,42)(26,27)(32,43)(37,47)(39,40)(41,48)

Polycyclic presentation:

2^9|a^2=e,b^2=egh,c^2=ef,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^c=de,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, 379696);