Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #162118 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2856 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2094 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,9,11,12)(8,10)(13,14,15,18,16,19,20,17);(5,7,8,11)(6,9)(10,12)(14,17)(15,20)(18,19);(1,2,3,4);(7,11)(9,12)Polycyclic presentation:
2^9|a^2=g,b^2=egi,c^2=f,e^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,g^b=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72314071, #72314085, #72314105 |
| maximal normal subgroup of | #72314071, #72314085, #72314105 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 112174);