#9712441

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

4046 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)(3,8)(4,5)(6,14)(7,9)(10,15)(11,13)(12,16)(17,18,23,21)(19,27,29,24)(20,30,31,25)(22,32,28,26)(33,34,39,43)(35,44,46,40)(36,41,47,37)(38,45,48,42);(1,3,7,12)(2,8,9,16)(4,13,15,6)(5,11,10,14)(17,19,23,29)(18,24,21,27)(20,28,31,22)(25,26,30,32)(33,35,39,46)(34,40,43,44)(36,45,47,42)(37,48,41,38);(2,9)(3,12)(4,15)(11,14)(17,20,23,31)(18,25,21,30)(19,22,29,28)(24,32,27,26)(33,36,39,47)(34,37,43,41)(35,45,46,42)(38,40,48,44);(1,4)(2,10)(3,13)(5,9)(6,12)(7,15)(8,14)(11,16)(17,20)(18,25)(19,28)(21,30)(22,29)(23,31)(24,26)(27,32)(33,36)(34,41)(35,45)(37,43)(38,40)(39,47)(42,46)(44,48);(1,5,7,10)(2,4,9,15)(3,14,12,11)(6,16,13,8)(17,21,23,18)(19,24,29,27)(20,30,31,25)(22,32,28,26)(33,37,39,41)(34,36,43,47)(35,38,46,48)(40,42,44,45);(1,6,7,13)(2,11,9,14)(3,4,12,15)(5,8,10,16)(17,22,23,28)(18,26,21,32)(19,20,29,31)(24,30,27,25)(33,38)(34,42)(35,37)(36,40)(39,48)(41,46)(43,45)(44,47)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=h,f^2=i,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, 9619637);