#480182

order	512
nilpotency class	3
isoclinism	isoclinic class of #1468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂³	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂×#121	C₈
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=g,f^2=h,h^2=i,b^a=be,e^a=eg,d^b=di,f^b=fg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387378);