#8720323

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3961 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)(4,8)(5,9)(7,11)(10,15)(12,14)(13,16)(17,18)(19,22)(20,30)(21,32)(23,27)(24,29)(25,31)(26,28)(35,41)(39,43)(42,47)(46,48);(1,3)(2,6)(4,10)(5,11)(7,9)(8,15)(12,16)(13,14)(17,19)(18,24)(20,21)(22,27)(23,29)(25,32)(26,30)(28,31)(33,34,38,37)(35,39,41,43)(36,44,45,40)(42,48,47,46);(1,2,5,9)(3,7,11,6)(4,8,14,12)(10,16,13,15)(17,20,23,31)(18,25,27,30)(19,21,29,28)(22,26,24,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,2)(3,7)(4,12)(5,9)(6,11)(8,14)(10,15)(13,16)(17,20)(18,25)(19,28)(21,29)(22,32)(23,31)(24,26)(27,30)(33,36,38,45)(34,40,37,44)(35,42,41,47)(39,46,43,48);(1,3)(2,7)(4,13)(5,11)(6,9)(8,15)(10,14)(12,16)(17,21,23,28)(18,26,27,32)(19,20,29,31)(22,25,24,30)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,4,5,14)(2,8,9,12)(3,10,11,13)(6,15,7,16)(17,22,23,24)(18,19,27,29)(20,26,31,32)(21,30,28,25)(33,37,38,34)(35,43,41,39)(36,44,45,40)(42,48,47,46)

Polycyclic presentation:

2^9|b^2=ghi,c^2=gi,d^2=ghi,e^2=g,f^2=h,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, 8627519);