Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| 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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1840 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)(7,8)(9,11)(10,12)(13,14,19,25,20,26,18,24)(15,21,28,37,17,34,35,23)(16,29,32,40,33,39,31,22)(27,38,41,44,30,43,42,36);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,20,17)(14,21,26,34)(16,27,33,30)(18,35,19,28)(22,36,40,44)(23,25,37,24)(29,38,39,43)(31,42,32,41);(2,4)(13,16,20,33)(14,22,26,40)(15,27,17,30)(18,32,19,31)(21,36,34,44)(23,38,37,43)(24,29,25,39)(28,42,35,41);(6,11)(7,12)(13,17,20,15)(14,23,26,37)(16,30,33,27)(18,35,19,28)(21,25,34,24)(22,38,40,43)(29,36,39,44)(31,42,32,41);(5,8,9,10)(6,7,11,12)(13,18,20,19)(14,24,26,25)(15,28,17,35)(16,31,33,32)(21,37,34,23)(22,39,40,29)(27,41,30,42)(36,38,44,43)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,e^c=ei,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411436);