Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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 |
| prime radical | #25106 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1909 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(13,14,18,23)(15,24,25,20)(16,21,26,28)(17,22,19,27);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,18,25)(14,20,23,24)(16,17,26,19)(21,27,28,22);(1,3,4,2)(6,9)(7,11)(13,16)(14,20)(15,19)(17,25)(18,26)(21,27)(22,28)(23,24);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,17)(18,26)(19,25)(20,27)(22,24)(23,28);(6,9)(10,12)(13,17,18,19)(14,22,23,27)(15,16,25,26)(20,28,24,21)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257105, #72257139, #72257176, … |
| maximal normal subgroup of | #72257105, #72257139, #72257176, … |
| maximal quotient of | #72308892 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6421679);