#6456189

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35210	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|c^2=fi,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363385);