#9379900

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

4396 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9287096);