Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#393 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2722 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,11)(8,18,20,12)(9,22)(10,13)(14,24)(15,31,32,25)(16,26)(17,34)(19,28,35,36)(21,29,23,30)(27,33);(1,2,3,4)(5,7)(6,11)(8,19,20,35)(9,17)(10,16)(12,28,18,36)(13,26)(14,27)(15,21,32,23)(22,34)(24,33)(25,29,31,30);(5,8)(6,12)(7,15)(9,23)(10,20)(11,25)(13,18)(14,30)(16,32)(17,35)(19,33)(21,24)(22,29)(26,31)(27,36)(28,34);(7,16)(11,26)(15,32)(17,33)(19,35)(25,31)(27,34)(28,36)Polycyclic presentation:
2^9|a^2=h,b^2=e,g^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^c=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72239324, #72239333, #72239429, … |
| maximal normal subgroup of | #72239324, #72239333, #72239429, … |
| maximal quotient of | #72279434, #72279808, #72303303, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264944);