#8633448

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

4099 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=gi,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, 8540644);