#8726622

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=h,e^2=i,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, 8633818);