Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437529 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(10,15)(11,18)(12,13)(17,20)(19,21)(22,24);(2,4)(5,6)(7,8)(9,10)(11,16)(12,15)(13,20)(14,17)(18,22)(19,23)(21,24);(6,8)(9,11)(10,16)(12,18)(13,21)(14,19)(15,22)(17,23)(20,24);(6,8)(9,12,14,13)(10,15,17,20)(11,18,19,21)(16,22,23,24);(10,17)(15,20)(16,23)(22,24)Polycyclic presentation:
2^9|d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72237165, #72249595, C2×#6437531, … |
| maximal normal subgroup of | #72237165, #72249595, C2×#6437531, … |
| maximal quotient of | #72237126, #72257675, #72257895, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344727);