Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
12132 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,19)(12,17)(13,18)(14,16)(15,20)(21,23)(22,24);(5,6)(7,8)(9,11)(10,15)(12,21)(13,17)(14,20)(16,19)(18,22)(23,24);(2,4)(10,16)(11,20)(17,24)(18,21);(1,2)(3,4)(9,11)(10,15)(12,21)(13,17)(14,20)(16,19)(18,22)(23,24);(9,12)(10,17)(11,21)(13,15)(14,22)(16,24)(18,20)(19,23);(9,13)(10,18)(11,17)(12,19)(14,23)(15,22)(16,21)(20,24)Polycyclic presentation:
2^9|a^2=gi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
| maximal subgroup of | #72234473, #72234475, #72234480, … |
| maximal normal subgroup of | #72234473, #72234475, #72234480, … |
| maximal quotient of | #72240165, #72252059, #72252064, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7603199);