#10047503

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

4118 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,23,24)(19,25,28,21)(20,22,30,32)(26,27,29,31)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,48)(45,46);(1,3,6,11)(2,7,8,16)(4,14,10,12)(5,13,9,15)(17,19,23,28)(18,21,24,25)(20,29,30,26)(22,27,32,31)(35,36)(39,42)(43,44)(47,48);(2,8)(3,11)(4,10)(13,15)(18,24)(20,30)(21,25)(26,29)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,4,6,10)(2,9,8,5)(3,12,11,14)(7,13,16,15)(17,20)(18,22)(19,26)(21,31)(23,30)(24,32)(25,27)(28,29)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,3)(2,7)(4,12)(5,15)(6,11)(8,16)(9,13)(10,14)(17,21)(18,19)(20,31)(22,26)(23,25)(24,28)(27,30)(29,32)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(1,5,6,9)(2,10,8,4)(3,13,11,15)(7,12,16,14)(17,22)(18,20)(19,27)(21,29)(23,32)(24,30)(25,26)(28,31)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9954699);