Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514000 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2281 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,10)(7,11,8,12)(13,14)(15,17)(16,24)(18,20)(19,22)(21,28)(23,27)(25,26);(1,2)(3,4)(6,10)(11,12)(13,15,17,14)(16,25,26,24)(18,22,19,20)(21,28,23,27);(14,20)(15,22)(18,19)(21,25)(23,24)(27,28);(2,4)(5,7,9,8)(6,11,10,12)(13,16,17,26)(14,21,15,23)(18,28,19,27)(20,25,22,24);(5,8,9,7)(6,11,10,12)(14,15)(20,22)(21,23)(24,25)Polycyclic presentation:
2^9|a^2=fi,b^2=i,d^2=f,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7429148);