Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2016 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,18,24)(15,25,28,20)(16,22,31,37)(17,23,19,33)(21,32,38,30)(26,40,43,35)(27,41,29,36)(34,42,39,44);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15,18,28)(14,20,24,25)(16,26,31,43)(17,27,19,29)(21,34,38,39)(22,35,37,40)(23,36,33,41)(30,42,32,44);(1,3,4,2)(7,11)(9,12)(13,16)(14,21)(15,26)(17,30)(18,31)(19,32)(20,34)(22,33)(23,37)(24,38)(25,39)(27,42)(28,43)(29,44)(35,41)(36,40);(7,11)(9,12)(13,16)(14,22)(15,26)(17,32)(18,31)(19,30)(20,35)(21,33)(23,38)(24,37)(25,40)(27,44)(28,43)(29,42)(34,41)(36,39);(6,10)(9,12)(13,17,18,19)(14,23,24,33)(15,27,28,29)(16,30,31,32)(20,36,25,41)(21,37,38,22)(26,42,43,44)(34,40,39,35)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377960);