#8622025

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

4162 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8529221);