Finite Group Databasebeta

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Cayley diagram

C2×#8300 #184737

order512
nilpotency class3
isoclinismisoclinic class of #1462
automorphism?
homologyAbelianization (H1) = C23×C8 · Schur multiplier (H2) = C29

Aliases

direct productC2×#8300

Subgroups

subgroupG/H
center, socleC24C2×C22⋊C4
commutatorC23C23×C8
Frattini, cosocleC23×C4C24
prime radicalC26C2×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

4110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 4 generators on 30 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72292811, #72292814
maximal normal subgroup of#72292811, #72292814
maximal quotient of#72292838, #72293116

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91933);