#476030

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

942 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)(17,18,23,28)(19,29,32,42)(20,33,34,41)(21,27,22,26)(24,37,39,31)(25,35,40,36)(30,44,45,38)(43,47,48,46);(2,5)(6,8)(9,10,12,14)(11,15,16,13)(20,34)(25,40)(30,45)(33,41)(35,36)(38,44)(43,48)(46,47);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,30)(21,37)(22,31)(23,32)(25,38)(26,42)(27,29)(28,39)(33,43)(34,45)(35,47)(36,46)(40,44)(41,48);(3,7)(6,8)(17,20,21,35,23,34,22,36)(18,25,26,41,28,40,27,33)(19,30,37,47,32,45,31,46)(24,38,42,48,39,44,29,43)

Polycyclic presentation:

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

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, 383226);