Finite Group Databasebeta

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Cayley diagram

C2×#24125 #6448568

order512
nilpotency class3
isoclinismisoclinic class of #23848
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#24125

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3726 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);(3,4)(5,6)(7,8)(9,13)(10,11)(12,14)(15,16)(17,23)(18,21)(19,22)(20,25)(24,29)(26,28)(27,30);(8,11)(12,13)(15,17,18,25)(16,20,21,23)(19,24,27,26)(22,28,30,29);(4,6)(17,24)(19,27)(20,28)(22,30)(23,29)(25,26);(7,9,10,14)(8,12,11,13)(16,21)(20,23)(22,30)(28,29)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72265079, #72265102, #72265107, …
maximal normal subgroup of#72265079, #72265102, #72265107, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355764);