Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| 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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2629 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)(13,14,17,20)(15,22,23,28)(16,19,26,21)(18,25,27,24);(1,2)(3,4)(5,7,9,6)(8,11,12,10);(2,4)(5,8,9,12)(6,10,7,11)(13,15)(14,18)(16,25)(17,23)(19,22)(20,27)(21,28)(24,26);(6,7)(10,11)(14,19)(16,26)(18,22)(20,21)(24,25)(27,28);(14,20)(15,23)(19,21)(24,25)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,c^b=ch,d^b=df,e^c=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308197, #72308260, #72308312 |
| maximal normal subgroup of | #72308197, #72308260, #72308312 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408972);