#9874919

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

4170 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)(5,7)(8,12)(9,15)(10,13)(14,16)(17,18)(19,21)(20,23)(22,28)(24,26)(25,30)(27,31)(29,32)(33,34,39,36)(35,40,37,41)(38,42,46,44)(43,47,45,48);(1,3,5,10)(2,6,7,13)(4,9,12,16)(8,14,11,15)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32)(33,34,39,36)(35,41,37,40)(38,42,46,44)(43,48,45,47);(2,7)(6,13)(8,11)(14,15)(18,20)(19,26)(25,28)(27,32)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,2)(3,6)(4,8)(5,7)(9,14)(10,13)(11,12)(15,16)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,36,39,34)(35,41,37,40)(38,44,46,42)(43,48,45,47);(1,3)(2,6)(4,9)(5,10)(7,13)(8,14)(11,15)(12,16)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,37,39,35)(34,41,36,40)(38,43,46,45)(42,47,44,48);(1,4)(2,8)(3,9)(5,12)(6,14)(7,11)(10,16)(13,15)(17,22,23,30)(18,25,20,28)(19,27,26,32)(21,29,24,31)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gi,d^2=i,e^2=gi,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, 9782115);