#9726379

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

4083 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)(3,8)(4,13)(5,10)(6,14)(7,9)(11,16)(12,15)(17,18)(19,25)(20,24)(21,26)(22,27)(23,30)(28,29)(31,32)(33,34)(35,36)(37,45)(38,42)(39,40)(41,47)(43,48)(44,46);(1,3,7,4)(2,8,9,13)(5,14,15,11)(6,12,16,10)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,27)(26,32)(33,35,38,40)(34,39,42,36)(37,43,47,46)(41,48,45,44);(2,9)(3,4)(10,12)(11,14)(33,36,38,39)(34,40,42,35)(37,44,47,48)(41,46,45,43);(1,4,7,3)(2,8,9,13)(5,14,15,11)(6,10,16,12)(17,20,22,26)(18,24,27,21)(19,28,30,32)(23,31,25,29)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,46)(47,48);(1,5,7,15)(2,10,9,12)(3,11,4,14)(6,8,16,13)(17,19)(18,25)(20,28)(21,31)(22,30)(23,27)(24,29)(26,32)(33,35)(34,36)(37,46)(38,40)(39,42)(41,48)(43,47)(44,45);(1,6,7,16)(2,11,9,14)(3,12,4,10)(5,13,15,8)(17,21,22,24)(18,26,27,20)(19,29,30,31)(23,32,25,28)(33,37)(34,41)(35,43)(36,44)(38,47)(39,48)(40,46)(42,45)

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=gh,e^2=i,f^2=gh,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, 9633575);