Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393988 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#392 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,13)(8,16)(9,12)(10,14)(11,18)(15,20)(17,19);(5,7)(6,10)(8,17)(9,15)(11,20)(12,19)(13,16)(14,18);(10,19)(13,20);(1,2,3,4)Polycyclic presentation:
2^9|d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72219661, #72227280, #72227283, … |
| maximal normal subgroup of | #72219661, #72227280, #72227283, … |
| maximal quotient of | #72237226, #72237228, #72237265, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318902);