#8840316

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	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

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, 8747512);