Finite Group Databasebeta

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Cayley diagram

C2×#28929 #6459974

order512
nilpotency class3
isoclinismisoclinic class of #24274
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#28929

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2608 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 256

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

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72265547, C2×#6465814, #72265600, …
maximal normal subgroup of#72265547, C2×#6465814, #72265600, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6367170);