Finite Group Databasebeta

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Cayley diagram

C22×#1867 #6371332

order512
nilpotency class3
isoclinismisoclinic class of B(4,𝔽2)
automorphism?
homologyAbelianization (H1) = C24×C4 · Schur multiplier (H2) = C211

Aliases

direct productC22×#1867

Subgroups

subgroupG/H
center, socleC24#120
commutatorC23C24×C4
Frattini, cosocleC24C25
prime radicalC24×D4C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C26 ◁ G
LECSG ▷ C24 ▷ C2 ▷ C1

7070 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 16
transitive degreeμt(G) = 128

A minimal faithful permutation representation — 5 generators on 16 points:

(1,2);(3,4);(5,6,8,7)(13,16);(5,7,8,6)(9,10)(11,13)(12,14)(15,16);(9,11)(10,13)(12,15)(14,16)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC22×#24404, C22×C4×B(4,𝔽2), #72278475, …
maximal normal subgroup ofC22×#24404, C22×C4×B(4,𝔽2), #72278475, …
maximal quotient ofC22×#10745, C2×#192094, C2×#192366, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278528);