Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C24×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C23 | C24×C4 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2388 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20);(5,7)(6,9)(8,12)(10,11)(13,15)(14,17)(16,19)(18,20)(21,22)(23,26)(24,25)(27,28);(5,7)(6,10)(8,12)(9,11)(14,18)(17,20)(22,25)(26,28);(7,12)(9,10)(15,19)(17,20);(21,23,24,27)(22,26,25,28)Polycyclic presentation:
2^9|a^2=i,e^2=h,b^a=bf,c^a=cg,c^b=ch,d^b=dfg,d^c=df104 conjugacy classes — view the full character table.
| maximal subgroup of | #72304039, #72304100, #72304123 |
| maximal normal subgroup of | #72304039, #72304100, #72304123 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 451508);