Finite Group Databasebeta

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Cayley diagram

C2×#28815 #6465743

order512
nilpotency class3
isoclinismisoclinic class of #24269
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28

Aliases

direct productC2×#28815

Subgroups

subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2034 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 512

A minimal faithful permutation representation — 5 generators on 30 points:

(1,2);(7,8,10,12)(9,13,14,11)(15,16)(17,25)(18,21)(19,29)(20,24)(22,27)(23,26)(28,30);(3,4,5,6)(15,17,26,28,20,27,19,18)(16,21,29,22,24,30,23,25);(4,6)(7,9,10,14)(8,11,12,13)(17,27)(18,28)(21,30)(22,25);(15,18,19,27,20,28,26,17)(16,22,23,21,24,25,29,30)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72265542, C2×#6465814, #72265594, …
maximal normal subgroup of#72265542, C2×#6465814, #72265594, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372939);