Finite Group Databasebeta

← Prev (#181615) Next (#181617) →

Cayley diagram

C2×#8052 #181616

order512
nilpotency class3
isoclinismisoclinic class of #1504
automorphism?
homologyAbelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42

Aliases

direct productC2×#8052

Subgroups

subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
socleC23C2×#95
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

1984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72239587, #72239601, #72239626, …
maximal normal subgroup of#72239587, #72239601, #72239626, …
maximal quotient of#72239594

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88812);