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 | |
2179 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)(13,14,18,24)(15,27,30,21)(16,33,29,43)(17,37,28,41)(19,26,20,25)(22,36,42,35)(23,38,44,34)(31,39,32,40);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,21)(16,34)(17,35)(18,30)(19,32)(20,31)(22,41)(23,43)(24,27)(25,40)(26,39)(28,36)(29,38)(33,44)(37,42);(13,16,18,29)(14,22,24,42)(15,28,30,17)(19,36,20,35)(21,23,27,44)(25,43,26,33)(31,34,32,38)(37,39,41,40);(2,4)(5,7,9,8)(6,11,12,10)(13,17,18,28)(14,23,24,44)(15,29,30,16)(19,38,20,34)(21,22,27,42)(25,41,26,37)(31,35,32,36)(33,39,43,40);(5,8,9,7)(6,11,12,10)(14,24)(15,30)(16,29)(23,44)(25,26)(31,32)(35,36)(37,41)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=f,e^2=fi,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, 7429896);