Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #15124 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1604 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)(15,18)(16,20)(17,19)(22,25)(27,28);(6,9)(10,12)(13,15,17,16)(14,18,19,20)(21,22,26,27,24,25,23,28);(5,7,8,11)(6,10,9,12)(14,19)(18,20)(22,27)(23,26)(25,28);(2,4)(7,11)(10,12)(13,16,17,15)(14,20,19,18)(21,23,24,26)(22,27,25,28)Polycyclic presentation:
2^9|b^2=fgi,c^2=ei,d^2=fi,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319974, #72320123, #72320369 |
| maximal normal subgroup of | #72319974, #72320123, #72320369 |
| maximal quotient of | #72321734 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 373245);