Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2028 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)(7,10)(8,9)(11,12)(13,14,18,21)(15,19,23,22)(16,20,27,17)(24,26,25,28);(6,9)(7,11)(13,15)(14,19)(16,26)(17,24)(18,23)(20,25)(21,22)(27,28);(1,2,3,4)(13,16,18,27)(14,20,21,17)(15,22,23,19)(24,26,25,28);(6,9)(10,12)(14,21)(15,23)(16,27)(24,25);(5,7,8,11)(6,10,9,12)(14,21)(15,24)(17,20)(19,28)(22,26)(23,25)Polycyclic presentation:
2^9|a^2=i,c^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283071, #72283158, #72284414, … |
| maximal normal subgroup of | #72283071, #72283158, #72284414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592389);