#13503

order	256
nilpotency class	4
isoclinism	isoclinic class of #13392
automorphism	Aut(G) = #72312248 (order 2048) · Inn(G) = #482 · Out(G) = C₂³×C₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#482
commutator, Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#2077
prime radical	C₂×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

136 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 136 conjugacy classes of subgroups — too many to draw. Only its 49 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13503, #499392, #499595, …
maximal normal subgroup of	C₂×#13503, #499392, #499595, …
maximal quotient of	#146043, #146047, #146073, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6491);