#10143445

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

4068 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,21,24)(19,25,27,22)(20,23,30,28)(26,32,29,31)(33,34,39,37)(35,40,45,47)(36,42,46,41)(38,48,44,43);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,18)(19,22)(20,28)(21,24)(23,30)(25,27)(26,32)(29,31)(33,35,39,45)(34,40,37,47)(36,44,46,38)(41,48,42,43);(2,3)(7,9)(8,10)(14,15)(17,19,21,27)(18,22,24,25)(20,26,30,29)(23,31,28,32)(33,36,39,46)(34,41,37,42)(35,38,45,44)(40,43,47,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19)(18,22)(20,29)(21,27)(23,32)(24,25)(26,30)(28,31)(33,36)(34,42)(35,44)(37,41)(38,45)(39,46)(40,43)(47,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,20)(18,23)(19,26)(21,30)(22,31)(24,28)(25,32)(27,29)(33,37,39,34)(35,40,45,47)(36,42,46,41)(38,43,44,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,18)(19,22)(20,28)(21,24)(23,30)(25,27)(26,32)(29,31)(33,38,39,44)(34,43,37,48)(35,36,45,46)(40,41,47,42)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gi,d^2=ghi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10050641);