Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 | #24439 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2253 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,21)(15,19,17,25)(16,23,24,27)(20,26,28,22);(1,2)(3,4)(6,9)(10,12)(13,15,18,17)(14,19,21,25)(16,22,24,26)(20,27,28,23);(2,4)(13,16,18,24)(14,20,21,28)(15,22,17,26)(19,27,25,23);(2,4);(5,7,8,11)(6,10,9,12)(16,24)(20,28)(22,26)(23,27)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,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=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72307099, #72307195, #72307267, … |
| maximal normal subgroup of | #72307099, #72307195, #72307267, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347072);