Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3554 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(6,9)(7,11)(13,14,19,24,18,23,17,22)(15,25,27,36,28,20,29,35)(16,30,31,21,32,37,33,38)(26,40,43,39,42,44,41,34);(5,6)(7,10)(8,9)(11,12)(13,15,18,28)(14,20,23,25)(16,26,32,42)(17,29,19,27)(21,34,38,39)(22,36,24,35)(30,44,37,40)(31,43,33,41);(1,2,3,4)(13,16)(14,21)(15,26)(17,31)(18,32)(19,33)(20,34)(22,37)(23,38)(24,30)(25,39)(27,41)(28,42)(29,43)(35,44)(36,40);(6,9)(10,12)(13,17,18,19)(14,22,23,24)(15,27,28,29)(16,31,32,33)(20,35,25,36)(21,37,38,30)(26,41,42,43)(34,44,39,40);(5,7,8,11)(6,10,9,12)(14,23)(15,28)(21,38)(22,24)(26,42)(27,29)(30,37)(41,43)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gh,d^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=df,e^b=ei,g^c=gi,h^c=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300956);