Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32935 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | #36624 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2600 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)(15,24)(16,20)(17,22)(18,27)(19,23)(21,26)(25,28);(6,9)(10,12)(13,15,18,16,19,26,17,25)(14,20,22,24,23,28,27,21);(5,7,8,11)(6,10,9,12)(13,16,18,26,19,25,17,15)(14,21,22,20,23,24,27,28);(2,4);(13,17,19,18)(14,22,23,27)(15,25,26,16)(20,24,28,21)Polycyclic presentation:
2^9|b^2=g,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ci104 conjugacy classes — view the full character table.
| maximal subgroup of | #72249392, #72249409, #72249470, … |
| maximal normal subgroup of | #72249392, #72249409, #72249470, … |
| maximal quotient of | #72308727, C2×#6629469 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536665);