Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #422715 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2428 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23438 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1837 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(2,4)(5,6,12,18)(7,19,24,38)(8,25,30,42)(9,31,35,45)(10,17,11,16)(13,23,39,22)(14,28,43,29)(15,33,46,34)(20,49,52,36)(21,53,55,37)(26,40,51,47)(27,58,60,41)(32,44,54,48)(50,63,68,65)(56,61,64,59)(57,62,67,66);(1,2,3,4)(5,7,11,23,12,24,10,22)(6,13,17,38,18,39,16,19)(8,26,29,20,30,51,28,52)(9,32,34,21,35,54,33,55)(14,40,25,36,43,47,42,49)(15,44,31,37,46,48,45,53)(27,50,61,57,60,68,59,67)(41,62,64,63,58,66,56,65);(5,8,10,29,12,30,11,28)(6,14,16,25,18,43,17,42)(7,20,22,51,24,52,23,26)(9,27,34,59,35,60,33,61)(13,36,19,47,39,49,38,40)(15,41,31,56,46,58,45,64)(21,50,54,67,55,68,32,57)(37,62,48,65,53,66,44,63);(2,4)(5,9)(6,15)(7,21)(8,27)(10,34)(11,33)(12,35)(13,37)(14,41)(16,31)(17,45)(18,46)(19,48)(20,50)(22,54)(23,32)(24,55)(25,56)(26,57)(28,61)(29,59)(30,60)(36,62)(38,44)(39,53)(40,63)(42,64)(43,58)(47,65)(49,66)(51,67)(52,68)Polycyclic presentation:
2^9|a^2=i,b^2=ei,c^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330094);