Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24278 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
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 | #60359 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2676 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)(7,13,15,10)(8,12)(9,11)(14,20,16,18)(17,19);(2,4)(7,14,15,16)(10,18,13,20);(5,7)(6,10)(8,16)(9,15)(11,13)(12,20)(14,17)(18,19);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216067, #72219654, #72219660, … |
| maximal normal subgroup of | #72216067, #72219654, #72219660, … |
| maximal quotient of | #72260721, #72261980, #72262054 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271628);