Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3151 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2727 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)(7,9)(8,10)(11,12)(13,14)(15,21)(16,27)(17,23)(18,22)(19,24)(20,25)(26,28);(1,2)(3,4)(5,7,9,6)(8,11,12,10);(2,4)(5,8,9,12)(6,10,7,11)(13,15,19,26)(14,20,24,27)(16,17,25,18)(21,23,28,22);(6,7)(10,11)(13,16)(14,21)(15,17)(18,26)(19,25)(20,23)(22,27)(24,28);(13,17,19,18)(14,22,24,23)(15,25,26,16)(20,21,27,28)Polycyclic presentation:
2^9|b^2=f,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72281515, #72282245, #72282644, … |
| maximal normal subgroup of | #72281515, #72282245, #72282644, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408923);