Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1197 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(14,19)(15,23)(16,17)(21,22)(24,25)(26,27);(1,3,4,2)(6,9)(10,11)(13,14)(15,20)(16,21)(17,19)(18,22)(23,26)(24,28)(25,27);(14,19)(16,17)(20,26)(21,22)(23,25)(27,28);(5,7,8,12)(6,10,9,11)(13,15)(14,20)(16,25)(17,23)(18,24)(19,26)(21,27)(22,28)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=efg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,g^c=gi,e^d=ei,f^d=fi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279815);