Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23998 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3508 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2934 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)(7,10,14,20)(8,11)(9,12)(13,17)(15,18,16,19);(2,4)(5,7)(6,10,12,20)(8,16)(9,14)(11,19,13,18)(15,17);(6,11)(8,17)(10,18)(12,13)(15,16)(19,20);(2,4)(5,7)(6,10,12,20)(8,15)(9,14)(11,18,13,19)(16,17)Polycyclic presentation:
2^9|a^2=h,b^2=g,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,f^c=fi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72219645, #72219657, #72219672, … |
| maximal normal subgroup of | #72219645, #72219657, #72219672, … |
| maximal quotient of | #72234064, #72234077, #72260837, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271168);