Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394108 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2433 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14807 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,12,8,13)(3,14,16,25,5,22,17,9)(4,18,20,28,21,10,23,27)(15,29,30,26,19,32,31,24);(1,3,8,16,7,5,6,17)(2,9,13,14,12,25,11,22)(4,15,23,30,21,19,20,31)(10,26,28,32,18,24,27,29);(1,4)(2,10)(3,15)(5,19)(6,20)(7,21)(8,23)(9,24)(11,27)(12,18)(13,28)(14,29)(16,30)(17,31)(22,32)(25,26);(1,5,8,17,7,3,6,16)(2,9,13,14,12,25,11,22)(4,19,23,31,21,15,20,30)(10,24,28,29,18,26,27,32)Polycyclic presentation:
2^9|a^2=e,b^2=ef,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,h^c=hi,e^d=ei,f^d=fi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72285493, #72285582, C2×#400797, … |
| maximal normal subgroup of | #72285493, #72285582, C2×#400797, … |
| maximal quotient of | C2×#400797 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 307993);