#8726508

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

4094 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)(3,11)(4,6)(5,8)(7,9)(10,15)(12,13)(14,16)(17,18)(19,26)(20,28)(21,23)(22,25)(24,29)(27,31)(30,32);(1,3,7,5)(2,8,9,11)(4,14,15,12)(6,13,10,16)(17,19,22,21)(18,23,25,26)(20,27,29,30)(24,31,28,32)(33,34)(35,38)(36,39)(37,40)(41,46)(42,47)(43,45)(44,48);(2,9)(4,15)(8,11)(12,14)(20,29)(24,28)(27,30)(31,32)(33,34,37,40)(35,38,42,47)(36,39,45,43)(41,46,44,48);(1,4,7,15)(2,10,9,6)(3,12,5,14)(8,13,11,16)(17,20)(18,24)(19,27)(21,30)(22,29)(23,31)(25,28)(26,32)(33,34)(35,38)(36,43)(37,40)(39,45)(41,48)(42,47)(44,46);(1,5,7,3)(2,11,9,8)(4,14,15,12)(6,13,10,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,31)(28,29)(30,32)(33,35,37,42)(34,38,40,47)(36,44,45,41)(39,48,43,46);(1,6,7,10)(2,4,9,15)(3,13,5,16)(8,14,11,12)(17,21,22,19)(18,23,25,26)(20,27,29,30)(24,32,28,31)(33,36,37,45)(34,39,40,43)(35,41,42,44)(38,46,47,48)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=h,e^2=hi,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, 8633704);