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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1828 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)(8,12)(9,10)(13,14)(15,27)(16,32)(17,36)(18,24)(19,26)(20,25)(21,29)(22,34)(23,38)(28,43)(30,40)(31,41)(33,39)(35,42)(37,44);(1,2)(3,4)(6,10)(11,12)(13,15,18,29)(14,21,24,27)(16,33,28,38)(17,34,37,35)(19,31,20,30)(22,36,42,44)(23,43,39,32)(25,41,26,40);(5,7,9,8)(6,11,10,12)(13,16)(14,22)(15,17)(18,28)(19,35)(20,34)(21,39)(23,27)(24,42)(25,43)(26,32)(29,37)(30,33)(31,38)(36,40)(41,44);(2,4)(13,17,18,37)(14,23,24,39)(15,28,29,16)(19,33,20,38)(21,22,27,42)(25,44,26,36)(30,34,31,35)(32,40,43,41);(5,8,9,7)(6,11,10,12)(14,24)(15,29)(16,28)(23,39)(25,26)(30,31)(34,35)(36,44)Polycyclic presentation:
2^9|b^2=i,c^2=fi,d^2=i,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, 7427402);