Finite Group Databasebeta

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Cayley diagram

C4×#3116 #6373304

order512
nilpotency class3
isoclinismisoclinic class of #3003
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C28

Aliases

direct productC4×#3116

Subgroups

subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

2348 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72283028
maximal normal subgroup of#72283028

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280500);