#9851370

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

4258 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9758566);