Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,7)(13,14,18,22)(15,23,24,27)(16,21,17,20)(19,26,28,25);(2,4)(9,10,11,12)(13,15)(14,19)(16,26)(17,25)(18,24)(20,27)(21,23)(22,28);(5,7,8,6)(14,20)(16,17)(19,27)(21,22)(23,28)(25,26);(15,24)(19,28)(23,27)(25,26)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gi,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72308457, #72308483, #72308494, … |
| maximal normal subgroup of | #72308457, #72308483, #72308494, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261143);