Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458129 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2420 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10,14,20,15,19,34,44)(11,16,23,38,24,37,53,64)(12,25,28,45,29,17,33,41)(13,30,31,59,32,18,42,43)(21,46,26,54,48,35,52,39)(22,49,50,70,51,36,62,63)(27,40,57,66,58,55,60,67)(47,61,56,65,69,68,71,72);(2,5)(6,8)(9,11,14,23,15,24,34,53)(10,16,20,38,19,37,44,64)(12,26,28,48,29,52,33,21)(13,22,31,50,32,51,42,62)(17,39,41,46,25,54,45,35)(18,36,43,63,30,49,59,70)(27,56,57,69,58,71,60,47)(40,65,66,68,55,72,67,61);(1,3)(2,6)(4,7)(5,8)(9,12,15,29)(10,17,19,25)(11,21,24,48)(13,27,32,58)(14,33,34,28)(16,35,37,46)(18,40,30,55)(20,45,44,41)(22,47,51,69)(23,52,53,26)(31,60,42,57)(36,61,49,68)(38,54,64,39)(43,67,59,66)(50,71,62,56)(63,72,70,65);(3,7)(6,8)(9,13)(10,18,19,30)(11,22)(12,27)(14,31)(15,32)(16,36,37,49)(17,40,25,55)(20,43,44,59)(21,47)(23,50)(24,51)(26,56)(28,57)(29,58)(33,60)(34,42)(35,61,46,68)(38,63,64,70)(39,65,54,72)(41,66,45,67)(48,69)(52,71)(53,62)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=i,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,g^c=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374679);