Finite Group Databasebeta

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Cayley diagram

C2×#23719 #6432356

order512
nilpotency class3
isoclinismisoclinic class of #23439
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#23719

Subgroups

subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2018C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2396 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)(5,8)(6,7)(9,10)(11,12,14,19,15,18,23,17)(13,16,20,25,22,24,21,26);(4,7)(8,10)(11,13)(12,16)(14,20)(15,22)(17,26)(18,24)(19,25)(21,23);(12,17)(13,20)(14,23)(18,19)(21,22)(25,26);(3,5,6,9)(4,8,7,10)(12,18)(16,24)(17,19)(25,26)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×#6467260, #72239146, #72322459
maximal normal subgroup ofC2×#6467260, #72239146, #72322459
maximal quotient of#72239146

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339552);