Finite Group Databasebeta

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Cayley diagram

C2×#21396 #6372390

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

Aliases

direct productC2×#21396

Subgroups

subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

3818 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)(7,8)(9,12)(10,11)(13,14)(15,16,20,22,19,23,18,24)(17,25,26,21,28,30,27,29);(3,4,5,6)(15,17)(16,21)(18,26)(19,28)(20,27)(22,25)(23,29)(24,30);(8,11)(12,14)(15,18,19,20)(16,22,23,24)(17,26,28,27)(21,25,29,30);(7,9,10,13)(8,12,11,14)(16,23)(21,29)(22,24)(25,30)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72264920, #72264935, #72264977
maximal normal subgroup of#72264920, #72264935, #72264977

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279586);