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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2570 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)(15,21)(16,22)(17,19)(18,20)(23,27)(24,26)(25,28);(1,2)(3,4)(6,9)(10,12)(14,19)(20,22)(21,26)(27,28);(2,4)(14,20)(16,18)(19,22)(21,27)(23,25)(26,28);(2,4)(13,15)(14,21)(16,25)(17,24)(18,23)(19,26)(20,27)(22,28);(5,7,8,11)(6,10,9,12)(13,16,17,18)(14,22,19,20)(15,23,24,25)(21,27,26,28)Polycyclic presentation:
2^9|e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^c=ei,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72239656, #72256825, #72256888, … |
| maximal normal subgroup of | #72239656, #72256825, #72256888, … |
| maximal quotient of | #72237269, #72264651, #72264752 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347076);