Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3713 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,22)(17,20)(18,21)(19,24)(25,27)(26,28);(2,4)(13,15)(14,19)(16,26)(17,25)(18,24)(20,28)(21,27)(22,23);(6,9)(10,12)(14,20)(15,24)(16,18)(19,27)(23,28)(25,26);(15,24)(16,18)(19,23)(21,22)(25,26)(27,28);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72263983, #72315243, #72319376, … |
| maximal normal subgroup of | #72263983, #72315243, #72319376, … |
| maximal quotient of | #72264160, #72316092, C4×#23848, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361103);