#10261531

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4403 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=gi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10168727);