Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401086 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14811 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7,13,22)(4,15)(5,18)(6,9)(8,17)(10,19)(11,23,14,20)(12,26,27,21)(16,24)(25,31,28,32)(29,30);(1,3)(2,7,9,22)(4,12)(5,11)(6,13)(8,21,15,26)(10,20,18,23)(14,19)(16,25)(17,27)(24,31,29,32)(28,30);(3,11)(5,19)(7,20)(10,18)(12,25)(13,14)(16,30)(21,31)(22,23)(24,29)(26,32)(27,28);(1,4)(2,8,9,15)(3,12)(5,16)(6,17)(7,21,22,26)(10,24,18,29)(11,25)(13,27)(14,28)(19,30)(20,31,23,32)Polycyclic presentation:
2^9|a^2=f,b^2=ef,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237642, #72240437, #72240441, … |
| maximal normal subgroup of | #72237642, #72240437, #72240441, … |
| maximal quotient of | #72237647, C2×#401086, #72255627, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308282);