Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2985 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,8,7);(6,7)(10,15)(11,19)(12,13)(17,18)(20,21)(22,23);(1,2,3,4)(5,6,8,7)(9,10,14,18)(11,16,20,24)(12,15,13,17)(19,22,21,23);(5,7,8,6)(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,23)(18,24);(9,12,14,13)(10,17,18,15)(11,19,20,21)(16,23,24,22)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72312721, #72312904, #72312915 |
| maximal normal subgroup of | #72312721, #72312904, #72312915 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351401);