Finite Group Databasebeta

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Cayley diagram

C4×#2397 #514040

order512
nilpotency class2
isoclinismisoclinic class of #392
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C210

Aliases

direct productC4×#2397

Subgroups

subgroupG/H
centerC22×C42C23
commutatorC23C24×C4
Frattini, socle, cosocleC24C25
prime radicalC2×#2397C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C42 ◁ G
LECSG ▷ C24 ▷ C1

5766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 256

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

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72211654, C2×C4×#2397, #72216614, …
maximal normal subgroup of#72211654, C2×C4×#2397, #72216614, …
maximal quotient of#72211630, C2×C4×#2397, #72239566

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 421236);