Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500908 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(6,11,14,13)(12,15,20,19);(1,2)(3,4)(5,6)(7,15)(8,11)(9,13)(10,14)(12,16)(17,20)(18,19);(2,4)(5,6)(7,12)(8,13)(9,11)(10,14)(15,16)(17,19)(18,20);(5,7)(6,12)(8,16)(9,17)(10,18)(11,19)(13,15)(14,20);(5,8,10,9)(6,13,14,11)(7,16,18,17)(12,15,20,19)Polycyclic presentation:
2^9|a^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,f^b=fi,g^c=gi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72227999, #72228005, #72228006 |
| maximal normal subgroup of | #72227999, #72228005, #72228006 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410570);