Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C25 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3122 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)(5,6,8,9)(7,11,12,10)(14,18)(15,20)(22,25)(23,27);(6,9)(10,11)(21,22,24,25)(23,28,27,26);(5,6)(7,10)(8,9)(11,12)(13,14,17,18)(15,19,20,16);(5,7)(6,10)(8,12)(9,11)(13,15)(14,19)(16,18)(17,20);(5,6)(7,10)(8,9)(11,12)(13,16,17,19)(14,15,18,20)(21,23,24,27)(22,26,25,28)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,e^b=ef128 conjugacy classes — view the full character table.
| maximal subgroup of | #72267633, #72267636 |
| maximal normal subgroup of | #72267633, #72267636 |
| maximal quotient of | #72267673 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 439122);