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Cayley diagram

C2×#15546 #609125

order512
nilpotency class2
isoclinismisoclinic class of #15257
automorphism?
homologyAbelianization (H1) = C25 · Schur multiplier (H2) = C27×C4

Aliases

direct productC2×#15546

Subgroups

subgroupG/H
center, socleC25C24
commutator, Frattini, cosocleC24C25
prime radicalC26C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C25 ◁ G
LECSG ▷ C24 ▷ C1

4410 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 256

A minimal faithful permutation representation — 5 generators on 30 points:

(1,2);(4,6)(8,11)(9,13)(15,16)(17,19)(18,21)(20,22)(23,24,26,27)(25,29,30,28);(16,17)(18,22)(24,27)(28,29);(7,8)(9,12)(10,11)(13,14)(15,17,19,16)(18,21,22,20)(23,25,26,30)(24,28,27,29);(3,4)(5,6)(7,9,10,13)(8,12,11,14)(15,18)(16,20)(17,21)(19,22)(23,24)(25,28)(26,27)(29,30)

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72329870, C2×#3019916, C2×#1470830
maximal normal subgroup of#72329870, C2×#3019916, C2×#1470830

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 516321);