#9712518

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

4039 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=g,d^2=h,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=di,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, 9619714);