#8889354

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3962 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)(17,18,21,22)(19,25,27,23)(20,28,29,24)(26,31,30,32)(33,34)(35,41)(36,42)(37,45)(38,39)(40,46)(43,44)(47,48);(1,3)(2,7)(4,8)(5,14)(6,9)(10,12)(11,15)(13,16)(18,22)(20,29)(23,25)(26,30)(34,39)(35,42)(40,45)(43,48);(2,8)(3,9)(10,13)(11,14)(17,19,21,27)(18,23,22,25)(20,26,29,30)(24,31,28,32)(33,35)(34,36)(37,43)(38,42)(39,41)(40,44)(45,47)(46,48);(1,3)(2,7)(4,8)(5,11)(6,9)(10,16)(12,13)(14,15)(17,19)(18,23)(20,30)(21,27)(22,25)(24,32)(26,29)(28,31)(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)(17,20,21,29)(18,24,22,28)(19,26,27,30)(23,31,25,32)(33,34)(35,36)(37,40)(38,39)(41,42)(43,44)(45,46)(47,48);(1,5,6,15)(2,10,8,13)(3,11,9,14)(4,12,7,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)(33,37,38,46)(34,40,39,45)(35,43,42,48)(36,44,41,47)

Polycyclic presentation:

2^9|a^2=gi,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, 8796550);