#9730628

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

4016 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=hi,d^2=g,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=di,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, 9637824);