Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518992 |
| 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 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2026 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)(7,10)(8,9)(11,12)(13,14)(15,23)(16,19)(17,20)(18,22)(21,26)(24,28)(25,27);(1,2)(3,4)(6,9)(10,12)(13,15,19,26)(14,21,16,23)(17,27,22,24)(18,25,20,28);(5,7,8,11)(6,10,9,12)(13,16,18,22,19,14,20,17)(15,24,28,23,26,27,25,21);(2,4)(13,17,18,16,19,22,20,14)(15,21,28,24,26,23,25,27);(6,9)(10,12)(13,18,19,20)(14,17,16,22)(15,25,26,28)(21,27,23,24)Polycyclic presentation:
2^9|b^2=i,c^2=fg,d^2=gi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72263899, #72263962, #72264022, … |
| maximal normal subgroup of | #72263899, #72263962, #72264022, … |
| maximal quotient of | #72264888, #72265130 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468334);