Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(7,11)(10,12)(13,14,18,20)(15,23,24,27)(16,21,26,25)(17,22,19,28);(2,3)(6,9)(7,11)(14,20)(16,26)(22,28)(23,27);(5,6)(7,10)(8,9)(11,12)(13,15,19,25,18,24,17,21)(14,16,22,23,20,26,28,27);(1,3,4,2)(6,9)(10,12)(13,16)(14,21)(15,22)(17,27)(18,26)(19,23)(20,25)(24,28);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,22,20,28)(15,21,24,25)(16,23,26,27)Polycyclic presentation:
2^9|a^2=f,c^2=g,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72314242 |
| maximal normal subgroup of | #72314242 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7063219);