#10040490

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

4114 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)(3,6)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,18,23,21)(19,27,29,24)(20,30,31,25)(22,32,28,26)(33,34,38,43)(35,41,44,39)(36,45,46,40)(37,47,48,42);(3,9)(6,13)(10,15)(14,16)(17,19,23,29)(18,24,21,27)(20,28,31,22)(25,26,30,32)(35,44)(36,46)(39,41)(40,45);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,20,23,31)(18,25,21,30)(19,22,29,28)(24,32,27,26)(33,35,38,44)(34,39,43,41)(36,47,46,42)(37,40,48,45);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20)(18,25)(19,28)(21,30)(22,29)(23,31)(24,26)(27,32)(33,36,38,46)(34,40,43,45)(35,42,44,47)(37,41,48,39);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,21,23,18)(19,24,29,27)(20,30,31,25)(22,32,28,26)(33,35)(34,41)(36,42)(37,45)(38,44)(39,43)(40,48)(46,47);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,22,23,28)(18,26,21,32)(19,20,29,31)(24,30,27,25)(33,37,38,48)(34,42,43,47)(35,40,44,45)(36,41,46,39)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9947686);