#10047459

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

4016 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=gi,e^2=g,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, 9954655);