Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #45093 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,23,19,24,17,20)(15,21,27,35,25,37,32,33)(16,22,28,39,31,40,30,34)(26,36,42,44,29,38,41,43);(5,6)(7,11)(8,10)(9,12)(14,20)(15,25)(16,28)(17,18)(21,35)(23,24)(26,41)(29,42)(30,31)(33,37)(34,39)(36,38);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,29)(17,32)(18,27)(19,25)(20,33)(22,38)(23,35)(24,37)(26,31)(28,41)(30,42)(34,44)(36,40)(39,43);(7,12)(9,11)(13,16,18,28,19,31,17,30)(14,22,23,39,24,40,20,34)(15,26,27,42,25,29,32,41)(21,36,35,44,37,38,33,43);(6,10)(9,11)(13,17,19,18)(14,23,24,20)(15,27,25,32)(16,30,31,28)(21,33,37,35)(22,39,40,34)(26,42,29,41)(36,43,38,44)Polycyclic presentation:
2^9|a^2=fg,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6913940);