Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460133 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16357 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1218 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)(3,7,14,23)(4,8)(5,18)(6,10)(9,17)(11,20)(12,27,13,21)(15,31,30,24)(16,25)(19,26)(22,28,32,29);(1,3)(2,7,10,23)(4,13)(5,15)(6,14)(8,21,25,27)(9,22,26,32)(11,24,18,31)(12,16)(17,29)(19,28)(20,30);(1,4)(2,8)(3,12,14,13)(5,19)(6,16)(7,21,23,27)(9,11)(10,25)(15,29,30,28)(17,20)(18,26)(22,31,32,24);(1,4)(2,9,10,26)(3,13)(5,17)(6,16)(7,22,23,32)(8,18,25,11)(12,14)(15,29)(19,20)(21,31,27,24)(28,30)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=f,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=de,g^c=gi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72285055, #72286066, #72286154, … |
| maximal normal subgroup of | #72285055, #72286066, #72286154, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367329);