#9206606

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

4100 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)(19,22)(20,23)(21,24)(25,26)(27,32)(28,31)(29,30)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,45)(46,47);(2,7)(4,13)(8,14)(10,12)(17,19)(18,22)(20,26)(21,29)(23,25)(24,30)(27,31)(28,32)(33,35,38,40)(34,37,41,42)(36,43,44,47)(39,46,45,48);(2,7)(3,11)(9,16)(10,12)(18,20)(19,25)(24,28)(27,29)(36,44)(39,45)(43,47)(46,48);(1,3)(2,8)(4,10)(5,9)(6,11)(7,14)(12,13)(15,16)(17,20,23,18)(19,26,25,22)(21,24,31,28)(27,32,29,30)(33,36,38,44)(34,39,41,45)(35,43,40,47)(37,46,42,48);(1,4,6,13)(2,9,7,16)(3,12,11,10)(5,8,15,14)(17,21)(18,24)(19,27)(20,28)(22,32)(23,31)(25,29)(26,30)(33,34)(35,37)(36,45)(38,41)(39,44)(40,42)(43,48)(46,47);(1,5,6,15)(2,10,7,12)(3,9,11,16)(4,14,13,8)(17,22,23,26)(18,25,20,19)(21,30,31,32)(24,27,28,29)(33,37)(34,40)(35,41)(36,46)(38,42)(39,47)(43,45)(44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,d^2=gh,e^2=hi,f^2=h,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, 9113802);