#8855126

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

4210 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:

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

Polycyclic presentation:

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

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, 8762322);