Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1881 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,23)(16,28)(17,19)(18,32)(20,27)(21,30)(22,25)(24,38)(26,41)(29,37)(31,36)(33,39)(34,44)(35,42)(40,43);(2,3)(5,7)(6,9)(8,12)(10,11)(14,19)(15,24)(21,28)(22,32)(26,29)(27,39)(36,43)(42,44);(1,2,4,3)(13,15,25,39,17,24,18,27)(14,20,32,38,19,33,22,23)(16,29,40,42,30,26,31,44)(21,35,43,37,28,34,36,41);(6,10)(9,11)(13,16,17,30)(14,21,19,28)(15,26,24,29)(18,31,25,40)(20,34,33,35)(22,36,32,43)(23,37,38,41)(27,42,39,44);(1,3,4,2)(7,12)(9,11)(15,24)(16,30)(20,33)(21,28)(23,38)(27,39)(31,40)(36,43)Polycyclic presentation:
2^9|c^2=gh,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6721951);