Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480263 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
3486 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)(7,15,17,28)(8,12)(9,21,23,13)(10,14)(11,18)(16,29,30,36)(19,32,34,26)(20,27)(22,25)(24,31)(33,35);(5,7)(6,11)(8,16)(9,22)(10,17)(12,24)(13,28)(14,25)(15,21)(18,23)(19,33)(20,30)(26,36)(27,35)(29,32)(31,34);(2,4)(5,8,10,20)(6,12,14,27)(7,16,17,30)(9,19,23,34)(11,24,25,35)(13,26,21,32)(15,29,28,36)(18,31,22,33);(7,17)(11,25)(15,28)(16,30)(18,22)(24,35)(29,36)(31,33)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=ei,g^b=ghi,g^e=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249343, #72249348, #72258026, … |
| maximal normal subgroup of | #72249343, #72249348, #72258026, … |
| maximal quotient of | #72249423, #72266189, #72266241, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400567);