Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #58990 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
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 | |
6376 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,8)(3,7,4,6)(9,10,13,12)(11,15,16,14)(17,18)(19,21)(20,22)(23,24);(1,3,5,4)(2,6,8,7)(9,11,13,16)(10,14,12,15)(19,23)(21,24);(10,12)(14,15)(17,19)(18,21)(20,23)(22,24);(1,3,5,4)(2,7,8,6)(9,12,13,10)(11,15,16,14);(18,22)(19,23);(1,4,5,3)(2,6,8,7)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^c=dh,e^c=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72239653, #72251184, #72256991, … |
| maximal normal subgroup of | #72239653, #72251184, #72256991, … |
| maximal quotient of | #72239781, #72265170, #72281487, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7573295);