#8750423

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3955 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)(20,26)(21,25)(22,28)(23,24)(29,32)(30,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,3)(2,7)(4,9)(5,12)(6,11)(8,16)(10,14)(13,15)(18,24)(20,30)(25,27)(28,29)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(3,11)(4,13)(7,14)(8,12)(17,19,23,21)(18,25,24,27)(20,28,30,29)(22,26,32,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,4,6,13)(2,8,10,12)(3,9,11,15)(5,7,16,14)(17,20,23,30)(18,26,24,31)(19,29,21,28)(22,27,32,25)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,2)(3,7)(4,8)(5,15)(6,10)(9,16)(11,14)(12,13)(17,21,23,19)(18,27,24,25)(20,28,30,29)(22,26,32,31)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47);(1,5,6,16)(2,9,10,15)(3,12,11,8)(4,14,13,7)(17,22,23,32)(18,28,24,29)(19,26,21,31)(20,25,30,27)

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=gi,e^2=g,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, 8657619);