Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | #30722 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1841 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)(15,26,17,21)(16,29,32,40)(18,24,20,35)(22,31,39,33)(23,34,37,28)(27,38,30,44)(36,42,41,43);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,19,17)(14,21,25,26)(16,27,32,30)(18,34,20,28)(22,36,39,41)(23,35,37,24)(29,44,40,38)(31,43,33,42);(2,4)(13,16,19,32)(14,22,25,39)(15,27,17,30)(18,33,20,31)(21,36,26,41)(23,38,37,44)(24,40,35,29)(28,43,34,42);(6,11)(7,12)(13,17,19,15)(14,23,25,37)(16,30,32,27)(18,34,20,28)(21,35,26,24)(22,38,39,44)(29,41,40,36)(31,43,33,42);(5,8,9,10)(6,7,11,12)(13,18,19,20)(14,24,25,35)(15,28,17,34)(16,31,32,33)(21,37,26,23)(22,29,39,40)(27,42,30,43)(36,38,41,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,e^b=ei,e^c=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410241);