#10188269

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

4089 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10095465);