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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2185 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)(7,11)(8,10)(9,12)(13,14,18,23)(15,25,29,20)(16,30,33,40)(17,24,19,21)(22,34,39,32)(26,37,28,36)(27,43,44,38)(31,41,42,35);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,20)(16,31)(17,26)(18,29)(19,28)(21,37)(22,38)(23,25)(24,36)(27,32)(30,35)(33,42)(34,44)(39,43)(40,41);(14,21)(15,26)(17,19)(20,25)(22,30)(23,24)(27,42)(28,29)(31,44)(32,34)(35,41)(39,40);(2,4)(5,7,9,8)(6,11,12,10)(13,16)(14,22)(15,27)(17,32)(18,33)(19,34)(20,35)(21,30)(23,39)(24,40)(25,41)(26,42)(28,31)(29,44)(36,38)(37,43);(5,8,9,7)(6,11,12,10)(13,17,18,19)(14,21,23,24)(15,28,29,26)(16,32,33,34)(20,36,25,37)(22,30,39,40)(27,31,44,42)(35,38,41,43)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=fi,e^2=f,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=cgi,d^b=dh,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,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, 7429893);