#8393421

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

4172 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)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,18)(19,28)(20,22)(21,31)(23,25)(24,32)(26,30)(27,29)(33,34)(35,39)(36,43)(37,38)(40,45)(41,48)(42,46)(44,47);(2,6)(3,10)(8,13)(11,16)(17,19,23,32)(18,24,25,28)(20,29,30,21)(22,31,26,27)(33,34)(35,39)(36,43)(37,38)(40,45)(41,48)(42,46)(44,47);(4,12)(8,13)(11,16)(14,15)(18,25)(22,26)(24,28)(27,31)(34,38)(39,46)(40,43)(47,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,20)(18,26)(19,29)(21,32)(22,25)(23,30)(24,27)(28,31)(33,35,37,42)(34,39,38,46)(36,44,45,41)(40,48,43,47);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,21,23,29)(18,27,25,31)(19,30,32,20)(22,24,26,28)(33,36,37,45)(34,40,38,43)(35,41,42,44)(39,47,46,48);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,22,23,26)(18,20,25,30)(19,31,32,27)(21,24,29,28)(33,34)(35,39)(36,40)(37,38)(41,47)(42,46)(43,45)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300617);