Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24420 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2362 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2,3,4)(5,6)(7,19)(8,25)(9,31)(10,17)(11,16)(12,18)(13,22)(14,29)(15,32)(20,35)(21,36)(23,38)(24,37)(26,53)(27,55)(28,41)(30,40)(33,43)(34,42)(39,57)(44,48)(45,49)(46,65)(47,60)(50,62)(51,61)(52,56)(54,64)(58,63)(59,66)(67,68);(5,7,12,24)(6,13,18,38)(8,26,30,20)(9,21,33,50)(10,23,11,22)(14,35,41,53)(15,36,42,62)(16,37,17,19)(25,44,40,60)(27,54,57,46)(28,47,29,48)(31,45,43,61)(32,51,34,49)(39,59,55,68)(52,64,63,65)(56,66,58,67);(2,4)(5,8,11,29,12,30,10,28)(6,14,17,25,18,41,16,40)(7,20,23,48,24,26,22,47)(9,27,32,56,33,57,34,58)(13,35,37,60,38,53,19,44)(15,39,31,52,42,55,43,63)(21,46,49,66,50,54,51,67)(36,59,61,65,62,68,45,64);(5,9)(6,15)(7,21)(8,27)(10,34)(11,32)(12,33)(13,36)(14,39)(16,43)(17,31)(18,42)(19,45)(20,46)(22,51)(23,49)(24,50)(25,52)(26,54)(28,58)(29,56)(30,57)(35,59)(37,61)(38,62)(40,63)(41,55)(44,64)(47,67)(48,66)(53,68)(60,65)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fh,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^c=gi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281301);