Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #24218 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1901 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)(6,10)(8,11)(14,19)(16,17)(20,22)(21,27)(23,28)(24,26);(1,2)(3,4)(5,6,9,10)(7,8,12,11)(14,20)(19,22)(21,23)(27,28);(5,7)(6,11)(8,10)(9,12)(13,14,17,19,18,20,16,22)(15,23,24,28,25,21,26,27);(2,4)(6,10)(7,12)(13,15)(14,21)(16,26)(17,24)(18,25)(19,27)(20,23)(22,28);(5,8,9,11)(6,7,10,12)(13,16,18,17)(14,22,20,19)(15,24,25,26)(21,27,23,28)Polycyclic presentation:
2^9|b^2=hi,c^2=g,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7092495);