#403063

order	512
nilpotency class	3
isoclinism	isoclinic class of #394289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

867 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(17,18)(19,29)(20,24)(21,35)(22,39)(23,28)(25,32)(26,37)(27,30)(31,40)(33,42)(34,41)(36,44)(38,43)(45,46)(47,48);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,30,34,23,32,22,20)(18,24,39,25,28,41,27,29)(21,31,37,46,38,47,36,33)(26,40,35,42,44,48,43,45);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,22,32,23,34,30,19)(18,25,27,24,28,29,39,41)(21,33,36,47,38,46,37,31)(26,42,43,40,44,45,35,48);(1,4,5,3)(2,8,6,7)(17,21,23,38)(18,26,28,44)(19,31,32,47)(20,33,34,46)(22,36,30,37)(24,40,41,48)(25,42,29,45)(27,43,39,35)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310259);