Finite Group Databasebeta

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Cayley diagram

C2×#24013 #6440308

order512
nilpotency class3
isoclinismisoclinic class of #23924
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C28×C4

Aliases

direct productC2×#24013

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

5108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 128

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

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72307703, #72307910, #72307954
maximal normal subgroup of#72307703, #72307910, #72307954

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347504);