#8889356

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	#55061	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3964 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,8)(3,7,9,4)(5,13,15,10)(11,12,14,16)(17,18,22,26)(19,27,30,23)(20,31,29,24)(21,32,28,25)(33,34)(35,41)(36,43)(37,45)(38,40)(39,46)(42,44)(47,48);(1,3)(2,7)(4,8)(5,14)(6,9)(10,12)(11,15)(13,16)(17,19,22,30)(18,23,26,27)(20,28,29,21)(24,25,31,32)(33,34,38,40)(35,41,43,36)(37,39,46,45)(42,47,48,44);(2,8)(3,9)(10,13)(11,14)(17,20,22,29)(18,24,26,31)(19,28,30,21)(23,25,27,32)(33,35)(34,36)(37,42)(38,43)(39,44)(40,41)(45,47)(46,48);(1,3)(2,7)(4,8)(5,11)(6,9)(10,16)(12,13)(14,15)(17,20)(18,24)(19,28)(21,30)(22,29)(23,25)(26,31)(27,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(1,4,6,7)(2,9,8,3)(5,12,15,16)(10,14,13,11)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,24,27,31)(33,34)(35,36)(37,39)(38,40)(41,43)(42,44)(45,46)(47,48);(1,5,6,15)(2,10,8,13)(3,11,9,14)(4,12,7,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,32)(29,31)(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=hi,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8796552);