Finite Group Databasebeta

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Cayley diagram

C2×#12854 #493855

order512
nilpotency class4
isoclinismisoclinic class of #11502
automorphism?
homologyAbelianization (H1) = C23×C4 · Schur multiplier (H2) = C27

Aliases

direct productC2×#12854

Subgroups

subgroupG/H
center, socleC23C2×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC24×D4C22
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25 ◁ C24×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

2528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 64

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72249386, #72249388, C2×#500059, …
maximal normal subgroup of#72249386, #72249388, C2×#500059, …
maximal quotient of#72263162

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401051);