Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 |
| prime radical | #23860 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,19,24)(15,20,16,22)(17,23,18,21)(25,28,26,27);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,19)(17,26)(18,25)(21,28)(22,24)(23,27);(2,4)(14,21)(15,16)(17,18)(20,27)(22,28)(23,24);(2,4)(13,16,19,15)(14,22,24,20)(17,25,18,26)(21,27,23,28);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,23,24,21)(15,25,16,26)(20,28,22,27)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^c=ei,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72283018, #72283706, #72284005, … |
| maximal normal subgroup of | #72283018, #72283706, #72284005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347479);