#10047181

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

3957 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)(3,7,11,5)(4,8,12,15)(10,13,16,14)(17,18)(19,27)(20,25)(21,28)(22,24)(23,30)(26,32)(29,31)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,48)(45,46);(1,3,6,11)(2,5,9,7)(4,10,12,16)(8,14,15,13)(17,19,22,30)(18,23,24,27)(20,26,31,28)(21,29,32,25)(35,36)(39,42)(43,44)(47,48);(4,12)(8,15)(10,16)(13,14)(18,24)(20,31)(23,27)(26,28)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,13)(12,16)(14,15)(17,20)(18,25)(19,28)(21,27)(22,31)(23,32)(24,29)(26,30)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,4)(2,8)(3,10)(5,14)(6,12)(7,13)(9,15)(11,16)(17,18)(19,27)(20,25)(21,28)(22,24)(23,30)(26,32)(29,31)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(1,5,6,7)(2,3,9,11)(4,13,12,14)(8,16,15,10)(17,21)(18,26)(19,29)(20,27)(22,32)(23,31)(24,28)(25,30)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,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, 9954377);