#10055717

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

4116 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)(3,7)(4,8)(5,6)(9,13)(10,15)(11,16)(12,14)(17,18,22,25)(19,20,23,26)(21,24,30,32)(27,28,31,29)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47);(2,6)(7,13)(8,14)(15,16)(17,19)(18,20)(21,27)(22,23)(24,28)(25,26)(29,32)(30,31)(33,34,37,40)(35,41,43,38)(36,39,46,44)(42,48,45,47);(3,9)(7,13)(10,11)(15,16)(19,23)(20,26)(27,31)(28,29)(33,34,37,40)(35,41,43,38)(36,39,46,44)(42,48,45,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,15)(11,16)(12,14)(17,18)(19,26)(20,23)(21,24)(22,25)(27,29)(28,31)(30,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,47)(44,46)(45,48);(1,3,5,9)(2,7,6,13)(4,11,12,10)(8,16,14,15)(17,20,22,26)(18,23,25,19)(21,29,30,28)(24,27,32,31)(33,35,37,43)(34,38,40,41)(36,45,46,42)(39,48,44,47);(1,4,5,12)(2,8,6,14)(3,10,9,11)(7,15,13,16)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,32)(26,29)(33,36,37,46)(34,39,40,44)(35,42,43,45)(38,47,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,e^2=gi,f^2=g,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, 9962913);