Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453994 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2670 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16351 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
963 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)(3,14,17,25)(4,9)(5,11)(6,12)(7,13)(8,16,10,18)(15,26,32,27)(19,24,20,31)(21,28)(22,30)(23,29);(1,3,5,18,7,17,6,16)(2,8,11,25,13,10,12,14)(4,19,21,15,22,20,23,32)(9,26,28,31,30,27,29,24);(1,4,5,21,7,22,6,23)(2,9,11,28,13,30,12,29)(3,15,16,19,17,32,18,20)(8,24,14,27,10,31,25,26);(1,3,5,18,7,17,6,16)(2,10,11,14,13,8,12,25)(4,20,21,32,22,19,23,15)(9,27,28,24,30,26,29,31)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=e,d^2=ef,e^2=i,g^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,g^b=gi,h^b=hi,d^c=de,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72237065, #72237068, #72237094, … |
| maximal normal subgroup of | #72237065, #72237068, #72237094, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367079);