Finite Group Databasebeta

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Cayley diagram

C2×#25781 #6443200

order512
nilpotency class3
isoclinismisoclinic class of #24264
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29×C4

Aliases

direct productC2×#25781

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24×D8C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

6242 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);(4,6)(11,12)(13,19)(14,23)(15,17)(16,21)(18,20)(22,25)(24,26);(7,8)(9,10)(11,13,20,26,15,22,14,21)(12,16,23,25,17,24,18,19);(3,4,5,6)(8,10)(12,17)(16,24)(18,23)(19,25);(11,13,20,26,15,22,14,21)(12,16,23,25,17,24,18,19)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72308104, #72308131, C2×#6573471, …
maximal normal subgroup of#72308104, #72308131, C2×#6573471, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350396);