#9706351

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

4037 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=gh,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,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, 9613547);