Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3114 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)(7,11)(8,10)(9,12)(13,14)(15,21)(16,18)(17,25)(19,23)(20,24)(22,28)(26,27);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(14,18)(19,27)(20,25)(21,28);(2,4)(13,15)(14,19)(16,22)(17,26)(18,27)(20,28)(21,25)(23,24);(6,10)(7,12)(14,20)(17,24)(18,25)(19,28)(21,27)(23,26);(15,22)(19,27)(21,28)(23,26)Polycyclic presentation:
2^9|g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,e^c=ei,g^c=gi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72312735, #72312809, #72312833 |
| maximal normal subgroup of | #72312735, #72312809, #72312833 |
| maximal quotient of | #72315868 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407891);