Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2232 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)(15,24)(16,29)(17,22)(18,21)(19,27)(20,32)(23,33)(25,37)(26,39)(28,36)(30,35)(31,40)(34,38)(41,42)(43,44);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,17,27)(14,19,22,24)(16,30,31,42)(18,26,33,28)(20,37,38,44)(21,36,23,39)(25,32,43,34)(29,41,40,35);(13,16)(14,20)(15,25)(17,31)(18,34)(19,35)(21,29)(22,38)(23,40)(24,41)(26,42)(27,43)(28,30)(32,33)(36,37)(39,44);(2,4)(5,7,9,8)(6,11,12,10)(14,21)(15,26)(18,33)(20,29)(22,23)(25,42)(27,28)(30,43)(32,34)(36,39)(37,44)(38,40);(5,8,9,7)(6,11,12,10)(14,22)(16,31)(19,24)(21,23)(25,43)(30,42)(32,34)(36,39)Polycyclic presentation:
2^9|b^2=f,d^2=fi,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7429806);