Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470894 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
2226 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)(7,12)(8,10)(9,11)(14,18)(16,19)(20,25)(21,26)(22,28)(24,27);(2,4)(7,8)(10,12);(5,7,9,8)(6,10,11,12)(13,14,19,26,17,21,16,18)(15,22,27,25,23,20,24,28);(5,8,9,7)(6,10,11,12)(13,15,17,23)(14,20,21,22)(16,24,19,27)(18,25,26,28);(6,11)(10,12)(14,21)(15,23)(18,26)(24,27)Polycyclic presentation:
2^9|c^2=gi,d^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72250360, #72250375, #72257039, … |
| maximal normal subgroup of | #72250360, #72250375, #72257039, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390369);