Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 |
| prime radical | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1817 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,12)(8,10)(9,11)(13,14,20,26)(15,21,31,42)(16,32,35,39)(17,36,33,38)(18,24,19,25)(22,34,43,28)(23,37,44,27)(29,40,30,41);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,20,31)(14,21,26,42)(16,33,35,17)(18,29,19,30)(22,23,43,44)(24,40,25,41)(27,34,37,28)(32,38,39,36);(5,8,9,7)(6,10,11,12)(13,16,20,35)(14,22,26,43)(15,27,31,37)(17,29,33,30)(18,34,19,28)(21,38,42,36)(23,40,44,41)(24,32,25,39);(2,4)(13,17,20,33)(14,23,26,44)(15,28,31,34)(16,29,35,30)(18,37,19,27)(21,39,42,32)(22,40,43,41)(24,36,25,38);(6,11)(10,12)(13,18,20,19)(14,24,26,25)(15,29,31,30)(16,34,35,28)(17,27,33,37)(21,40,42,41)(22,32,43,39)(23,38,44,36)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=fi,d^2=i,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7422735);