#10068864

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4342 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)(19,25)(20,24)(21,26)(22,23)(27,32)(28,29)(30,31)(35,41)(39,42)(43,47)(46,48);(1,2)(3,6)(4,9)(5,8)(7,11)(10,14)(12,16)(13,15)(17,19)(18,21)(20,28)(22,26)(23,25)(24,32)(27,30)(29,31)(33,34,38,36)(35,42,41,39)(37,40,45,44)(43,48,47,46);(2,3)(4,11)(8,10)(12,13)(18,23)(19,26)(24,31)(27,28)(34,36)(39,42)(40,44)(46,48);(1,3,6,2)(4,9,11,7)(5,10,14,8)(12,16,13,15)(17,19)(18,21)(20,27)(22,26)(23,25)(24,29)(28,30)(31,32)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,4,6,11)(2,7,3,9)(5,13,14,12)(8,16,10,15)(17,20,22,30)(18,24,23,31)(19,27,26,28)(21,29,25,32)(33,36,38,34)(35,42,41,39)(37,40,45,44)(43,46,47,48);(1,5)(2,8)(3,10)(4,12)(6,14)(7,15)(9,16)(11,13)(17,21)(18,19)(20,29)(22,25)(23,26)(24,27)(28,31)(30,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)

Polycyclic presentation:

2^9|b^2=hi,d^2=gi,e^2=i,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, 9976060);