Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1738 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,7)(9,10,12,11)(13,14)(15,23)(16,22)(17,21)(18,20)(19,26)(24,28)(25,27);(5,6,8,7)(13,15)(14,19)(16,25)(17,26)(18,24)(20,27)(21,23)(22,28);(2,4)(5,6,8,7)(14,20)(15,24)(16,18)(21,22)(25,26)(27,28);(5,7,8,6)(9,11,12,10)(13,16,17,18)(14,20,21,22)(15,25,26,24)(19,27,23,28)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=ei,d^2=eh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264651, #72264675, #72264693, … |
| maximal normal subgroup of | #72264651, #72264675, #72264693, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283484);