#10196230

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

4074 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=i,d^2=h,e^2=g,f^2=i,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, 10103426);