Finite Group Databasebeta

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Cayley diagram

C4×#2404 #514816

order512
nilpotency class2
isoclinismisoclinic class of #2403
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C211

Aliases

direct productC4×#2404

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, socle, cosocleC24C25
prime radicalC2×#2404C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

7266 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 20 points:

(1,2,3,4);(5,6)(7,8)(13,14)(15,19)(16,18)(17,20);(9,10)(11,12)(13,15,16,20)(14,17,18,19);(6,8)(10,12);(14,18)(17,19)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72232037, #72232043, #72232062, …
maximal normal subgroup of#72232037, #72232043, #72232062, …
maximal quotient ofC4×#23996, #72264638, AGL(1,𝔽5)×#2404, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 422012);