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Cayley diagram

C2×#25639 #6449212

order512
nilpotency class3
isoclinismisoclinic class of #23860
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#25639

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#496C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3870 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);(4,6)(7,8)(9,13)(10,11)(12,14)(15,16)(17,26)(18,28)(19,25)(20,24)(21,23)(22,27)(29,30);(8,11)(12,13)(15,17,20,27)(16,22,24,26)(18,21,29,19)(23,28,25,30);(3,4,5,6)(15,18,21,27,20,29,19,17)(16,22,25,28,24,26,23,30);(7,9,10,14)(8,12,11,13)(15,19,20,21)(16,23,24,25)(17,18,27,29)(22,28,26,30)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72282981
maximal normal subgroup of#72282981

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356408);