#8393637

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

4161 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)(19,27)(20,29)(21,26)(22,23)(24,30)(25,31)(28,32);(2,6)(3,10)(8,13)(11,16)(18,23)(20,31)(24,27)(26,28)(33,34,38,39)(35,41,44,40)(36,45,47,37)(42,48,46,43);(4,12)(8,13)(11,16)(14,15)(18,23)(21,32)(24,27)(25,29)(34,39)(37,45)(40,41)(43,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,19,22,30)(18,24,23,27)(20,26,31,28)(21,29,32,25)(33,35,38,44)(34,40,39,41)(36,46,47,42)(37,43,45,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,20,22,31)(18,25,23,29)(19,28,30,26)(21,24,32,27)(33,36)(34,37)(35,42)(38,47)(39,45)(40,48)(41,43)(44,46);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,21,22,32)(18,26,23,28)(19,29,30,25)(20,24,31,27)(33,37)(34,36)(35,43)(38,45)(39,47)(40,46)(41,42)(44,48)

Polycyclic presentation:

2^9|b^2=i,d^2=hi,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, 8300833);