Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #45236 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1883 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)(5,6)(7,9)(8,10)(11,12)(13,14,20,26)(15,27,30,42)(16,31,28,22)(17,23,32,43)(18,25,19,24)(21,37,40,29)(33,39,34,44)(35,38,36,41);(5,6)(7,11)(8,10)(9,12)(13,15,19,29,20,30,18,37)(14,21,25,27,26,40,24,42)(16,32,33,35,28,17,34,36)(22,41,39,23,31,38,44,43);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,20,28)(14,22,26,31)(15,17,30,32)(18,34,19,33)(21,38,40,41)(23,27,43,42)(24,44,25,39)(29,36,37,35);(7,12)(9,11)(13,17,20,32)(14,23,26,43)(15,28,30,16)(18,36,19,35)(21,39,40,44)(22,42,31,27)(24,38,25,41)(29,33,37,34);(6,10)(9,11)(13,18,20,19)(14,24,26,25)(15,29,30,37)(16,33,28,34)(17,35,32,36)(21,27,40,42)(22,39,31,44)(23,41,43,38)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,e^d=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910394);