#10055551

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) = 128

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=g,e^2=gi,f^2=g,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, 9962747);