Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,11,12,10)(13,14,17,18)(15,19,22,26)(16,23,25,20)(21,24,28,27);(5,6)(7,10)(8,9)(11,12)(13,15,17,22)(14,19,18,26)(16,24,25,27)(20,21,23,28);(1,2,3,4)(6,9)(10,11)(13,16,17,25)(14,20,18,23)(15,21,22,28)(19,27,26,24);(5,7,8,12)(6,10,9,11)(16,25)(20,23)(21,28)(24,27);(14,18)(15,19)(20,23)(21,27)(22,26)(24,28)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72282981, #72283071, #72283529, … |
| maximal normal subgroup of | #72282981, #72283071, #72283529, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591968);