#9711940

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

4085 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=gh,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 9619136);