Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471133 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16761 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
662 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,11,7,9,6,12)(3,8,13,24,15,23,16,21)(4,17,18,25,19,26,20,10)(14,28,29,22,30,31,27,32);(1,3,6,13,7,15,5,16)(2,8,11,21,9,23,12,24)(4,14,18,27,19,30,20,29)(10,22,26,31,25,32,17,28);(2,9)(3,13)(5,6)(8,21)(14,29)(15,16)(17,26)(18,20)(22,31)(23,24)(27,30)(28,32);(1,4)(2,10,9,25)(3,14)(5,20)(6,18)(7,19)(8,22,23,32)(11,26,12,17)(13,27)(15,30)(16,29)(21,31,24,28)Polycyclic presentation:
2^9|a^2=efg,b^2=fh,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,d^c=de,g^c=gi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251688, #72251692, #72251701, … |
| maximal normal subgroup of | #72251688, #72251692, #72251701, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378329);