Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11692 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| socle | C22 | C22×#99 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6,7,8)(9,10,13,18,14,17,25,31)(11,19,22,32,21,15,26,28)(12,16,23,30,24,29,36,39)(20,33,35,40,34,27,37,38);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,27)(17,19)(18,32)(23,35)(24,34)(25,26)(28,31)(29,33)(30,40)(36,37)(38,39);(1,3,5,4)(2,6,7,8)(9,12,14,24)(10,16,17,29)(11,20,21,34)(13,23,25,36)(15,27,19,33)(18,30,31,39)(22,35,26,37)(28,38,32,40);(11,21)(15,19)(20,34)(22,26)(27,33)(28,32)(35,37)(38,40)Polycyclic presentation:
2^9|a^2=f,c^2=hi,e^2=h,f^2=i,b^a=be,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,f^e=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72266604 |
| maximal normal subgroup of | #72266604 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388736);