Finite Group Databasebeta

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Cayley diagram

C4×#2756 #521270

order512
nilpotency class2
isoclinismisoclinic class of #2668
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4

Aliases

direct productC4×#2756

Subgroups

subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

2894 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)(7,9)(8,11)(10,12)(13,14)(15,19)(16,18)(17,20)(21,22,24,26)(23,25,27,28);(5,7)(6,8)(9,12)(10,11)(21,23)(22,25)(24,27)(26,28);(5,8,10,9)(6,11,12,7)(13,15,16,20)(14,17,18,19)(23,27)(25,28);(5,9,10,8)(6,11,12,7)(14,18)(17,19)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72303964, #72304046, #72304102
maximal normal subgroup of#72303964, #72304046, #72304102

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 428466);