Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #156684 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C25 | C24 |
| prime radical | C26 | C23 |
| LCS | G ▷ C25 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C1 | |
3268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(6,7)(21,22,24,26)(23,25,27,28);(5,6,8,7)(10,11)(15,19)(17,20)(23,27)(25,28);(9,10,12,11)(13,14,16,18)(15,17,19,20)(21,22)(23,25)(24,26)(27,28);(1,2,3,4)(5,7,8,6)(9,11,12,10)(13,15)(14,17)(16,19)(18,20)(21,23,24,27)(22,25,26,28)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=fi,d^2=fgh,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72328436, #72328645, #72328991, … |
| maximal normal subgroup of | #72328436, #72328645, #72328991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 65978);