#8720415

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

3946 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=hi,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, 8627611);