#10047500

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

4328 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)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18)(19,23)(20,21)(22,24)(25,26)(27,32)(28,31)(29,30)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,48)(45,46);(1,2,5,8)(3,6,9,13)(4,7,12,16)(10,14,11,15)(17,19,22,27)(18,23,24,32)(20,30,26,28)(21,29,25,31)(35,36)(39,42)(43,44)(47,48);(3,9)(6,13)(10,11)(14,15)(18,24)(19,27)(20,26)(29,31)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,20,22,26)(18,25,24,21)(19,28,27,30)(23,29,32,31)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19)(18,23)(20,28)(21,31)(22,27)(24,32)(25,29)(26,30)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(17,21,22,25)(18,26,24,20)(19,29,27,31)(23,28,32,30)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=hi,d^2=i,f^2=gh,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, 9954696);