#9026659

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	#41712	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,d^2=gh,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8933855);