Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2011 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,12)(7,11,8,10)(13,14,17,21)(15,18,23,26)(16,19,25,20)(22,27,24,28);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,18)(16,22)(17,23)(19,27)(20,28)(21,26)(24,25);(14,19)(15,22)(16,25)(20,21)(23,24)(27,28);(2,4)(14,19)(15,22)(16,25)(20,21)(23,24)(27,28);(5,8,9,7)(6,11,12,10)(15,23)(18,26)(22,24)(27,28)Polycyclic presentation:
2^9|a^2=fi,b^2=f,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72279952, #72282866, #72283105, … |
| maximal normal subgroup of | #72279952, #72282866, #72283105, … |
| maximal quotient of | #72311602, #72311719 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426627);