Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2766 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,19,18)(15,20,26,25)(16,21,29,33)(17,22,32,31)(23,34,39,27)(24,35,42,41)(28,36,30,37)(38,44,40,43);(5,7)(6,9)(8,12)(10,11)(13,15,19,26)(14,20,18,25)(16,27,29,34)(17,24,32,42)(21,23,33,39)(22,35,31,41)(28,43,30,44)(36,38,37,40);(2,4)(13,16,19,29)(14,21,18,33)(15,23,26,39)(17,30,32,28)(20,34,25,27)(22,37,31,36)(24,40,42,38)(35,43,41,44);(6,10)(9,11)(13,17,19,32)(14,22,18,31)(15,24,26,42)(16,28,29,30)(20,35,25,41)(21,36,33,37)(23,38,39,40)(27,43,34,44);(7,12)(9,11)(14,18)(15,25)(16,29)(20,26)(22,31)(23,34)(24,41)(27,39)(28,30)(35,42)(38,44)(40,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6574846);