#9668304

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

3891 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,d^2=ghi,e^2=ghi,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, 9575500);