Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24802 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #60444 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1989 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,10,14,19)(8,12)(9,11)(13,17,15,20)(16,18);(2,4)(7,13,14,15)(10,17,19,20);(5,7,8,15,9,14,16,13)(6,10,18,17,11,19,12,20);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216067, #72219660, #72219672, … |
| maximal normal subgroup of | #72216067, #72219660, #72219672, … |
| maximal quotient of | #72262054 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275974);