#10166801

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

4228 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=hi,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=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, 10073997);