#9668332

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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=i,d^2=hi,e^2=hi,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,e^d=egi,f^d=fgi

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, 9575528);