Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6912 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)(9,10)(11,19)(12,15)(13,18)(14,17)(16,21)(20,23)(22,24);(2,4)(5,6)(7,8)(10,15)(11,20)(12,13)(16,19)(17,18)(23,24);(6,8)(9,11,14,20)(10,16,17,24)(12,21,13,22)(15,23,18,19);(6,8)(9,12,14,13)(10,15,17,18)(11,21,20,22)(16,23,24,19);(10,17)(15,18)(16,24)(19,23)Polycyclic presentation:
2^9|c^2=i,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi,g^c=gi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72231709, #72233821, #72234091, … |
| maximal normal subgroup of | #72231709, #72233821, #72234091, … |
| maximal quotient of | #72233812, #72249810, #72257849, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344712);