Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8392 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(6,9)(7,11)(13,14)(15,20)(16,24)(17,18)(19,21)(22,23)(25,26)(27,28);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15,19,16)(14,20,21,24)(17,22,27,25)(18,23,28,26);(5,6)(7,10)(8,9)(11,12)(14,21)(18,28)(20,24)(23,26);(5,7,8,11)(6,10,9,12)(13,16,19,15)(14,20,21,24)(17,22,27,25)(18,26,28,23);(13,17)(14,18)(15,22)(16,25)(19,27)(20,23)(21,28)(24,26);(13,18)(14,17)(15,23)(16,26)(19,28)(20,22)(21,27)(24,25)Polycyclic presentation:
2^9|a^2=gi,b^2=h,d^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^c=fhi,e^d=ehi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72249482, #72249502, #72249524, … |
| maximal normal subgroup of | #72249482, #72249502, #72249524, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7556352);