#9236454

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
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3750 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=i,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9143650);