Finite Group Databasebeta

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Cayley diagram

C2×#32970 #6629104

order512
nilpotency class3
isoclinismisoclinic class of #32888
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#32970

Subgroups

subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×2−1+4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

4078 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);(3,4,6,8)(5,9,10,7)(12,16)(14,21)(17,25)(18,22)(19,23)(24,26);(3,5,6,10)(4,7,8,9);(11,12)(13,17)(14,22)(15,18)(16,21)(19,24)(20,26)(23,25);(11,13,15,20)(12,17,18,26)(14,23,21,19)(16,24,22,25)

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72271523, #72271527, #72271566
maximal normal subgroup of#72271523, #72271527, #72271566

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536300);