#8726266

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4027 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)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(33,34)(35,44)(36,37)(38,42)(39,43)(40,47)(41,46)(45,48);(2,3)(7,9)(8,10)(15,16)(17,18,22,23)(19,24,26,20)(21,25,31,32)(27,30,29,28)(33,35)(34,40)(36,45)(37,42)(38,46)(39,47)(41,48)(43,44);(2,3)(7,9)(8,10)(15,16)(18,23)(19,26)(25,32)(27,29)(33,36,39,46)(34,37,43,41)(35,45,47,38)(40,42,44,48);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,19)(18,24)(20,23)(21,29)(22,26)(25,28)(27,31)(30,32)(33,36)(34,41)(35,45)(37,43)(38,47)(39,46)(40,48)(42,44);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20,22,24)(18,19,23,26)(21,30,31,28)(25,29,32,27)(33,37,39,41)(34,36,43,46)(35,42,47,48)(38,40,45,44);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21,22,31)(18,25,23,32)(19,27,26,29)(20,28,24,30)(33,38,39,45)(34,42,43,48)(35,46,47,36)(37,44,41,40)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8633462);