#9203833

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

4090 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9111029);