Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #42648 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1831 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(6,9)(7,11)(13,14,19,25)(15,21,29,38)(16,30,33,22)(17,23,20,26)(18,24,28,37)(27,43,44,36)(31,42,35,39)(32,41,34,40);(5,6)(7,10)(8,9)(11,12)(13,15,19,29)(14,21,25,38)(16,31,33,35)(17,28,20,18)(22,39,30,42)(23,37,26,24)(27,32,44,34)(36,40,43,41);(2,4)(13,16,17,32,19,33,20,34)(14,22,23,40,25,30,26,41)(15,27,18,35,29,44,28,31)(21,36,24,42,38,43,37,39);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,23,25,26)(15,28,29,18)(16,32,33,34)(21,37,38,24)(22,40,30,41)(27,31,44,35)(36,39,43,42);(6,9)(10,12)(13,18,20,15,19,28,17,29)(14,24,26,21,25,37,23,38)(16,31,34,44,33,35,32,27)(22,39,41,43,30,42,40,36)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=hi,d^2=fi,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6715393);