Finite Group Databasebeta

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Cayley diagram

C2×#24202 #6448800

order512
nilpotency class3
isoclinismisoclinic class of #23860
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C29

Aliases

direct productC2×#24202

Subgroups

subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3664 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)(17,25)(18,19)(21,29)(22,23)(26,30)(27,28);(8,11)(12,13)(15,16,20,24)(17,26,27,29)(18,23,19,22)(21,25,30,28);(4,6)(15,17,20,27)(16,21,24,30)(18,28,19,25)(22,26,23,29);(7,9,10,14)(8,12,11,13)(17,27)(21,30)(25,28)(26,29)

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72264445, C2×#7613946, #72264567, …
maximal normal subgroup of#72264445, C2×#7613946, #72264567, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6355996);