Finite Group Databasebeta

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Cayley diagram

C22×#2840 #7633355

order512
nilpotency class2
isoclinismisoclinic class of #2812
automorphism?
homologyAbelianization (H1) = C26 · Schur multiplier (H2) = C212×C42

Aliases

direct productC22×#2840

Subgroups

subgroupG/H
center, socleC25C24
commutator, Frattini, cosocleC23C26
prime radicalC27C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C23 ▷ C1

15034 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 28
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72227570, #72227874
Fitting subgroup of#72227570, #72227602, C22×#73292
maximal subgroup of#72219195, #72219206, #72227570, …
maximal normal subgroup of#72219195, #72219206, #72227570, …
maximal quotient of#72219811, #72240918, C2×#6631461, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540551);