Finite Group Databasebeta

← Prev (#357577) Next (#357579) →

Cayley diagram

C2×#11020 #357578

order512
nilpotency class3
isoclinismisoclinic class of #2018
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C29

Aliases

direct productC2×#11020

Subgroups

subgroupG/H
center, socleC24#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×AGL(1,ℤ/8)C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

4958 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)(5,6)(7,8)(9,10)(11,12,15,19,16,17,14,22)(13,18,20,25,21,24,23,26);(4,6)(12,17)(13,20)(14,15)(18,25)(21,23)(24,26);(8,10)(11,13)(12,18)(14,23)(15,20)(16,21)(17,24)(19,25)(22,26)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#357581
maximal normal subgroup ofC2×#357581

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264774);