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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1819 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)(15,26,16,21)(17,31,33,39)(18,24,20,35)(22,28,37,30)(23,34,40,32)(27,38,29,44)(36,43,41,42);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,19,16)(14,21,25,26)(17,27,33,29)(18,30,20,28)(22,24,37,35)(23,36,40,41)(31,44,39,38)(32,43,34,42);(2,4)(13,16,19,15)(14,22,25,37)(17,29,33,27)(18,30,20,28)(21,24,26,35)(23,38,40,44)(31,41,39,36)(32,43,34,42);(6,11)(7,12)(13,17,19,33)(14,23,25,40)(15,27,16,29)(18,34,20,32)(21,36,26,41)(22,38,37,44)(24,31,35,39)(28,43,30,42);(5,8,9,10)(6,7,11,12)(13,18,19,20)(14,24,25,35)(15,28,16,30)(17,32,33,34)(21,22,26,37)(23,39,40,31)(27,42,29,43)(36,44,41,38)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,g^c=gi,e^d=ei,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410209);