Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | #30447 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1891 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,10)(8,11)(13,14,18,21)(15,19,20,28)(16,24,27,23)(17,25,26,22);(1,2)(3,4)(5,6,9,10)(7,8,12,11)(13,15)(14,19)(16,25)(17,23)(18,20)(21,28)(22,27)(24,26);(5,7)(6,11)(8,10)(9,12)(13,16,26,28,18,27,17,19)(14,20,25,23,21,15,22,24);(2,4)(6,10)(7,12)(13,15)(14,19)(16,25)(17,23)(18,20)(21,28)(22,27)(24,26);(5,8,9,11)(6,7,10,12)(14,21)(15,20)(22,25)(23,24)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=gi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7092476);