Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #470944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16720 |
| 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 | |
731 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)(3,13)(4,17,19,27)(5,11)(6,10)(7,12)(8,15)(9,20,28,21)(14,30,31,26)(16,25)(18,23,32,29)(22,24);(1,3)(2,8,12,13)(4,18,19,32)(5,22)(6,16)(7,15)(9,26)(10,24,11,25)(14,21,31,20)(17,23)(27,29)(28,30);(1,4,5,21,7,19,6,20)(2,9,11,27,12,28,10,17)(3,14,22,18,15,31,16,32)(8,23,25,26,13,29,24,30);(2,10)(4,19)(5,6)(8,24)(9,27)(11,12)(13,25)(16,22)(17,28)(18,32)(23,30)(26,29)Polycyclic presentation:
2^9|a^2=efg,b^2=efh,c^2=ef,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,f^b=fi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251689, #72251693, #72251701, … |
| maximal normal subgroup of | #72251689, #72251693, #72251701, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378140);