Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #425083 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2576 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15538 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
991 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,14,8,13,6,12)(3,15,19,9,20,30,18,29)(4,21,23,32,17,10,25,28)(5,24,26,31,16,11,22,27);(1,3)(2,9)(4,22)(5,25)(6,19)(7,18)(8,20)(10,24)(11,21)(12,30)(13,29)(14,15)(16,23)(17,26)(27,28)(31,32);(1,4)(2,10)(3,16)(5,20)(6,23)(7,25)(8,17)(9,27)(11,30)(12,28)(13,21)(14,32)(15,24)(18,22)(19,26)(29,31);(1,5)(2,11,13,24)(3,17)(4,20)(6,22)(7,26)(8,16)(9,28,29,32)(10,30,21,15)(12,31,14,27)(18,23)(19,25)Polycyclic presentation:
2^9|a^2=g,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72338481, #72338485, #72338500 |
| maximal normal subgroup of | #72338481, #72338485, #72338500 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 342978);