Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| 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×#3551 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3276 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)(5,6)(7,11)(8,10)(9,12)(13,14,18,24,19,25,17,23)(15,20,27,35,29,37,28,36)(16,22,30,21,32,38,31,39)(26,34,41,33,42,43,40,44);(5,7)(6,9)(8,12)(10,11)(13,15,19,29)(14,20,25,37)(16,26,32,42)(17,28,18,27)(21,33,39,44)(22,34,38,43)(23,36,24,35)(30,41,31,40);(1,2,3,4)(13,16)(14,21)(15,26)(17,30)(18,31)(19,32)(20,33)(22,24)(23,38)(25,39)(27,40)(28,41)(29,42)(34,35)(36,43)(37,44);(7,12)(9,11)(13,16)(14,22)(15,26)(17,31)(18,30)(19,32)(20,34)(21,24)(23,39)(25,38)(27,41)(28,40)(29,42)(33,35)(36,44)(37,43);(6,10)(9,11)(13,17,19,18)(14,23,25,24)(15,27,29,28)(16,30,32,31)(20,35,37,36)(21,38,39,22)(26,40,42,41)(33,34,44,43)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=gh,e^2=i,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=hi,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300628);