Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393975 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3987 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,13)(9,14)(10,24)(11,16)(12,20)(15,26)(18,33)(19,35)(21,32)(22,31)(23,30)(25,28)(27,36)(29,34);(5,7)(6,12)(8,18)(9,19)(10,25)(11,21)(13,29)(14,27)(15,32)(16,28)(17,24)(20,26)(22,36)(23,34)(30,35)(31,33);(2,4)(5,8,11,9)(6,13,16,14)(7,18,21,19)(10,23,26,22)(12,27,28,29)(15,31,24,30)(17,33,32,35)(20,36,25,34);(5,9,11,8)(6,14,16,13)(7,19,21,18)(10,22,26,23)(12,27,28,29)(15,30,24,31)(17,33,32,35)(20,34,25,36)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249727, #72255937, #72256289, … |
| maximal normal subgroup of | #72249727, #72255937, #72256289, … |
| maximal quotient of | #72252111, #72302154, C2×#393978 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301174);