Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2097 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)(7,9)(8,10)(11,12)(13,14,19,28,18,22,17,21)(15,16,23,25,20,27,24,26);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,18,20)(14,16,22,27)(17,24,19,23)(21,26,28,25);(2,4)(5,8,9,12)(6,10,7,11)(13,16)(14,20)(15,22)(17,25)(18,27)(19,26)(21,24)(23,28);(6,7)(10,11)(13,16)(14,20)(15,22)(17,25)(18,27)(19,26)(21,24)(23,28);(13,17,18,19)(14,21,22,28)(15,23,20,24)(16,25,27,26)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410900);