#9206759

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

4014 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)(3,10,5,8)(4,9,12,13)(11,16,14,15)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,34)(35,39)(36,37)(38,41)(40,44)(42,43)(45,48)(46,47);(1,3,7,5)(2,8,6,10)(4,11,12,14)(9,15,13,16)(17,19,23,22)(18,24,20,26)(21,29,30,27)(25,32,28,31)(33,35)(34,39)(36,42)(37,43)(38,44)(40,41)(45,46)(47,48);(4,12)(9,13)(11,14)(15,16)(18,20)(24,26)(25,28)(31,32)(33,35,38,44)(34,39,41,40)(36,42,46,45)(37,43,47,48);(1,4)(2,9)(3,11)(5,14)(6,13)(7,12)(8,15)(10,16)(17,20,23,18)(19,26,22,24)(21,25,30,28)(27,31,29,32)(33,35)(34,40)(36,45)(37,43)(38,44)(39,41)(42,46)(47,48);(1,5,7,3)(2,8,6,10)(4,11,12,14)(9,16,13,15)(17,21,23,30)(18,25,20,28)(19,27,22,29)(24,31,26,32)(33,36,38,46)(34,37,41,47)(35,42,44,45)(39,43,40,48);(1,6,7,2)(3,10,5,8)(4,13,12,9)(11,16,14,15)(17,22,23,19)(18,26,20,24)(21,27,30,29)(25,31,28,32)(33,37)(34,36)(35,43)(38,47)(39,42)(40,45)(41,46)(44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9113955);