Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1304 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)(13,14,16,17)(15,18,19,20);(6,9)(10,12)(14,17)(18,20)(21,22,25,26,24,27,23,28);(5,7,8,11)(6,10,9,12)(14,17)(15,19)(22,26)(23,25)(27,28);(2,4)(7,11)(10,12)(13,15,16,19)(14,18,17,20)(22,27)(26,28)Polycyclic presentation:
2^9|a^2=f,b^2=gi,c^2=ef,d^2=f,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 | #72320003, #72320085, #72320369 |
| maximal normal subgroup of | #72320003, #72320085, #72320369 |
| maximal quotient of | #72320933 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 374149);