Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2100 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)(5,6)(7,10)(8,11)(9,12)(13,14,18,22)(15,20,24,27)(16,21,17,26)(19,25,28,23);(1,2,3,4)(7,8)(10,11);(5,7,9,8)(6,10,12,11)(13,15)(14,19)(16,23)(17,25)(18,24)(20,26)(21,27)(22,28);(5,8,9,7)(6,11,12,10)(13,15)(14,20)(16,25)(17,23)(18,24)(19,26)(21,28)(22,27);(6,12)(10,11)(13,16,18,17)(14,21,22,26)(15,23,24,25)(19,27,28,20)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=h,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394928);