Finite Group Databasebeta

← Prev (#187954) Next (#187956) →

Cayley diagram

C2×#8538 #187955

order512
nilpotency class3
isoclinismisoclinic class of #8534
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C28

Aliases

direct productC2×#8538

Subgroups

subgroupG/H
center, socleC23#379
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#3505C2
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

2632 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);(7,10)(11,12,15,20,16,19,14,18)(13,21,22,26,23,17,24,25);(3,4)(5,7)(6,8)(9,10)(11,13)(12,17)(14,22)(15,24)(16,23)(18,25)(19,21)(20,26);(3,5)(4,7)(6,9)(8,10)(11,14,16,15)(12,18,19,20)(13,22,23,24)(17,25,21,26)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#187956
maximal normal subgroup ofC2×#187956

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95151);