Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500186 |
| 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 | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14,18,22,17,19,30,33)(15,23,26,32,25,37,31,20)(16,27,29,43,28,21,35,36)(24,39,41,34,40,38,44,42);(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(14,19)(20,32)(21,27)(22,33)(23,37)(34,42)(36,43)(38,39);(5,7)(6,9)(8,12)(10,11)(13,15,17,25)(14,20,19,32)(16,24,28,40)(18,31,30,26)(21,34,27,42)(22,37,33,23)(29,44,35,41)(36,39,43,38);(7,12)(9,11)(13,16,17,28)(14,21,19,27)(15,24,25,40)(18,29,30,35)(20,34,32,42)(22,36,33,43)(23,38,37,39)(26,41,31,44);(6,10)(9,11)(16,28)(21,27)(24,40)(29,35)(34,42)(36,43)(38,39)(41,44)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6410058);