#13438

order	256
nilpotency class	4
isoclinism	isoclinic class of #13436
automorphism	Aut(G) = #72317013 (order 4096) · Inn(G) = #2044 · Out(G) = C₂²×D₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#2044
commutator, Frattini, prime radical, cosocle	C₂×#41	C₂³
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

203 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 203 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,8,13)(3,14,19,39)(4,20,25,53)(5,26,7,11)(6,28,30,12)(9,18,34,16)(10,35,38,23)(15,44,47,64)(17,49,29,43)(21,54,45,60)(22,52,24,51)(27,40,31,42)(32,55,61,62)(33,48,41,50)(36,59,63,46)(37,56,58,57);(1,3,8,19)(2,9,13,34)(4,21,25,45)(5,18,7,16)(6,29,30,17)(10,36,38,63)(11,39,26,14)(12,41,28,33)(15,22,47,24)(20,44,53,64)(23,55,35,62)(27,48,31,50)(32,58,61,37)(40,43,42,49)(46,56,59,57)(51,60,52,54);(1,4,5,22,8,25,7,24)(2,10,11,37,13,38,26,58)(3,15,16,45,19,47,18,21)(6,23,27,56,30,35,31,57)(9,32,14,36,34,61,39,63)(12,20,40,51,28,53,42,52)(17,46,48,62,29,59,50,55)(33,60,43,44,41,54,49,64)

Polycyclic presentation:

2^8|a^2=h,b^2=h,c^2=d,d^2=h,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fgh,c^b=cf,d^b=dh,e^b=eg,f^b=fh,g^c=gh,f^e=fh

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#151242, #151243, #151266, …
maximal normal subgroup of	#151242, #151243, #151266, …
maximal quotient of	#144677, #144713, #144744, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6426);