#9508933

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

4030 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18)(19,27)(20,29)(21,26)(22,30)(23,25)(24,28)(31,32)(33,34,39,36)(35,40,43,44)(37,41,47,45)(38,46,48,42);(2,6)(7,9)(8,12)(14,15)(17,19,23,28)(18,24,25,27)(20,21,30,31)(22,26,29,32)(33,35)(34,40)(36,44)(37,46)(38,45)(39,43)(41,48)(42,47);(2,6)(3,10)(4,13)(14,15)(18,25)(19,28)(20,30)(26,32)(34,36)(35,43)(37,47)(38,48);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,20)(18,22)(19,21)(23,30)(24,26)(25,29)(27,32)(28,31)(33,36,39,34)(35,44,43,40)(37,41,47,45)(38,46,48,42);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,21)(18,26)(19,20)(22,24)(23,31)(25,32)(27,29)(28,30)(33,37)(34,41)(35,42)(36,45)(38,40)(39,47)(43,46)(44,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22)(18,20)(19,26)(21,24)(23,29)(25,30)(27,31)(28,32)(33,38,39,48)(34,42,36,46)(35,45,43,41)(37,40,47,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9416129);