Finite Group Databasebeta

← Prev (#6451756) Next (#6451758) →

Cayley diagram

C2×#23955 #6451757

order512
nilpotency class3
isoclinismisoclinic class of #23876
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C210

Aliases

direct productC2×#23955

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#3494C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

4118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72317964, C2×#6499844, #72318128
maximal normal subgroup of#72317964, C2×#6499844, #72318128

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6358953);