#8228523

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

4122 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8135719);