Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11370 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24);(1,2,3,4)(5,6,8,7);(6,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23);(10,15)(14,17)(16,22)(21,23);(5,7,8,6)(11,18)(14,17)(19,24)(21,23);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)Polycyclic presentation:
2^9|b^2=gi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,e^c=ehi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72234482, #72234484, #72234843, … |
| maximal normal subgroup of | #72234482, #72234484, #72234843, … |
| maximal quotient of | #72256871, #72256972, #72264072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7543934);