#10040457

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3805 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18)(19,22)(20,28)(21,24)(23,30)(25,27)(26,32)(29,31)(33,34,39,40)(35,41,45,48)(36,42,46,37)(38,43,47,44);(1,3,6,11)(2,7,8,16)(4,14,10,12)(5,13,9,15)(17,18,21,24)(19,25,27,22)(20,23,30,28)(26,32,29,31)(34,40)(36,46)(41,48)(43,44);(2,8)(3,11)(4,10)(13,15)(17,18,21,24)(19,25,27,22)(20,23,30,28)(26,32,29,31)(33,35,39,45)(34,41,40,48)(36,44,46,43)(37,47,42,38);(1,4,6,10)(2,9,8,5)(3,12,11,14)(7,13,16,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,31)(28,30)(29,32)(33,36)(34,42)(35,44)(37,40)(38,41)(39,46)(43,45)(47,48);(1,3)(2,7)(4,12)(5,15)(6,11)(8,16)(9,13)(10,14)(17,19,21,27)(18,22,24,25)(20,29,30,26)(23,32,28,31)(33,37,39,42)(34,36,40,46)(35,38,45,47)(41,43,48,44);(1,5,6,9)(2,10,8,4)(3,13,11,15)(7,12,16,14)(17,20,21,30)(18,23,24,28)(19,26,27,29)(22,31,25,32)(33,38)(34,43)(35,37)(36,41)(39,47)(40,44)(42,45)(46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=i,e^2=g,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, 9947653);