Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#447 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1131 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,9,11,8,7,10,12)(13,14,18,22)(15,20,24,28)(16,21,17,19)(23,26,25,27);(14,19)(16,17)(20,26)(21,22)(23,25)(27,28);(2,4)(6,7)(11,12)(13,15)(14,20)(16,25)(17,23)(18,24)(19,26)(21,27)(22,28);(5,7,9,12,8,6,10,11)(13,16,18,17)(14,21,22,19)(15,23,24,25)(20,26,28,27)Polycyclic presentation:
2^9|a^2=g,d^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=eh,d^c=dhi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72318702, #72319475, #72319529, … |
| maximal normal subgroup of | #72318702, #72319475, #72319529, … |
| maximal quotient of | #72321344, #72321438, #72322715 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 116428);