Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2730 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)(7,11)(8,10)(9,12)(13,14,18,22)(15,23,25,36)(16,20,27,34)(17,29,31,21)(19,30,33,26)(24,37,38,44)(28,41,40,35)(32,42,43,39);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(16,27)(20,34)(24,38)(28,40)(32,43)(35,41)(37,44)(39,42);(2,4)(13,15,18,25)(14,19,22,33)(16,24,27,38)(17,30,31,26)(20,32,34,43)(21,36,29,23)(28,42,40,39)(35,44,41,37);(6,10)(9,11)(13,16)(14,20)(15,24)(17,28)(18,27)(19,32)(21,35)(22,34)(23,37)(25,38)(26,39)(29,41)(30,42)(31,40)(33,43)(36,44);(2,4)(7,12)(9,11)(15,25)(16,27)(19,33)(20,34)(23,36)(26,30)(28,40)(35,41)Polycyclic presentation:
2^9|a^2=i,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6575292);