#8750500

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 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)(3,7)(4,12)(5,9)(6,10)(8,13)(11,14)(15,16)(17,18)(19,27)(20,21)(22,24)(23,26)(25,31)(28,29)(30,32)(33,34)(35,36)(37,41)(38,39)(40,44)(42,43)(45,46)(47,48);(1,3)(2,7)(4,9)(5,12)(6,11)(8,16)(10,14)(13,15)(17,19,23,22)(18,24,26,27)(20,30,25,28)(21,29,31,32)(34,39)(35,40)(41,46)(42,47);(3,11)(4,13)(7,14)(8,12)(17,19,23,22)(18,24,26,27)(20,30,25,28)(21,29,31,32)(33,35,38,40)(34,36,39,44)(37,42,45,47)(41,43,46,48);(1,4,6,13)(2,8,10,12)(3,9,11,15)(5,7,16,14)(17,20,23,25)(18,21,26,31)(19,28,22,30)(24,32,27,29)(33,36)(34,40)(35,39)(37,43)(38,44)(41,47)(42,46)(45,48);(1,2)(3,7)(4,8)(5,15)(6,10)(9,16)(11,14)(12,13)(17,21)(18,25)(19,29)(20,26)(22,32)(23,31)(24,28)(27,30)(33,37,38,45)(34,41,39,46)(35,42,40,47)(36,43,44,48);(1,5,6,16)(2,9,10,15)(3,12,11,8)(4,14,13,7)(17,22,23,19)(18,24,26,27)(20,28,25,30)(21,29,31,32)

Polycyclic presentation:

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

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, 8657696);