Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(6,9)(7,11)(13,14)(15,20)(16,22)(17,24)(18,23)(19,25)(21,26)(27,28);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,19)(17,27)(18,26)(21,23)(22,25)(24,28);(2,4)(13,15,19,16)(14,21,25,28)(17,26,18,27)(20,24,22,23);(5,7,8,11)(6,10,9,12)(13,16,19,15)(14,22,25,20)(17,26,18,27)(21,23,28,24);(2,4)(6,9)(10,12)(13,17,19,18)(14,23,25,24)(15,26,16,27)(20,28,22,21)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72282968, #72283838, #72284228, … |
| maximal normal subgroup of | #72282968, #72283838, #72284228, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6579151);