Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1530 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)(7,10)(8,9)(11,12)(13,14,16,18)(15,17,19,20)(22,25)(23,28)(26,27);(13,15)(14,17)(16,19)(18,20)(21,22)(23,27)(24,26)(25,28);(6,9)(10,12)(15,19)(17,20)(22,26)(25,27);(2,4)(5,7,8,11)(6,10,9,12)(22,27)(23,28)(25,26)Polycyclic presentation:
2^9|a^2=gi,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239705, #72239722, #72239728, … |
| maximal normal subgroup of | #72239705, #72239722, #72239728, … |
| maximal quotient of | #72257189, #72257314 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342998);