#10042494

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4044 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9949690);