Finite Group Databasebeta
Cayley diagram

C2×#14830 #607788

order512
nilpotency class2
isoclinismisoclinic class of #14715
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#14830

Subgroups

subgroupG/H
center, socleC25C24
commutator, Frattini, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

4878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 26 points:

(1,2);(4,6)(8,9)(11,12,14,16)(13,15,17,18)(19,20)(21,22)(23,24)(25,26);(7,8,10,9)(13,17)(15,18)(20,24)(21,26);(11,13)(12,15)(14,17)(16,18)(19,21)(20,25)(22,24)(23,26);(3,4)(5,6)(7,9,10,8)(11,12)(13,15)(14,16)(17,18)(19,22,23,25)(20,21,24,26)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#1104678, C2×#1431280, C2×#1265297, …
maximal normal subgroup ofC2×#1104678, C2×#1431280, C2×#1265297, …
maximal quotient ofC2×#158342, C2×#157314, C2×#154233

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 514984);