Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7357 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,20)(15,18,22,26)(16,19,24,28)(21,25,23,27);(6,9)(10,12)(13,15)(14,18)(16,23)(17,22)(19,27)(20,26)(21,24)(25,28);(1,2)(3,4)(13,16)(14,19)(15,21)(17,24)(18,25)(20,28)(22,23)(26,27);(2,4)(15,22)(18,26)(21,23)(25,27);(14,20)(16,24)(18,26)(21,23);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=i,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,d^c=dh,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72238144, #72238155, #72238183, … |
| maximal normal subgroup of | #72238144, #72238155, #72238183, … |
| maximal quotient of | #72263476, #72303388, #72313238, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7574889);