Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #459299 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2752 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16338 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1178 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)(3,10)(4,16)(5,8)(6,21)(7,12)(9,18)(11,14)(13,29)(15,24)(17,32)(19,28)(20,27)(22,25)(23,31)(26,30);(1,3,6,20,7,5,14,15)(2,8,11,27,12,10,21,24)(4,17,18,30,19,13,22,31)(9,23,16,29,25,26,28,32);(1,4)(2,9)(3,13)(5,17)(6,22)(7,19)(8,23)(10,26)(11,28)(12,25)(14,18)(15,31)(16,21)(20,30)(24,29)(27,32);(1,5,6,15,7,3,14,20)(2,10,11,24,12,8,21,27)(4,13,18,31,19,17,22,30)(9,23,16,29,25,26,28,32)Polycyclic presentation:
2^9|b^2=ef,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,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, #72237067, #72237071, … |
| maximal normal subgroup of | #72237065, #72237067, #72237071, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366495);