#13311

order	256
nilpotency class	4
isoclinism	isoclinic class of #13310
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#1999
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

260 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 57 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)(3,15)(4,21)(5,25)(6,14)(7,12)(8,13)(9,17)(10,23)(11,28)(16,45)(18,38)(19,43)(20,44)(22,54)(24,41)(26,32)(27,40)(29,39)(30,37)(31,55)(33,50)(34,49)(35,60)(36,57)(42,47)(46,52)(48,62)(51,61)(53,56)(58,63)(59,64);(1,3,8,20)(2,9,13,33)(4,16,24,48)(5,26,29,49)(6,19,7,18)(10,35,37,31)(11,38,40,43)(12,34,14,32)(15,39,44,25)(17,27,50,28)(21,52,41,59)(22,55,57,60)(23,51,30,47)(36,62,54,45)(42,53,61,63)(46,58,64,56);(1,4)(2,10)(3,16)(5,22)(6,30)(7,23)(8,24)(9,31)(11,36)(12,41)(13,37)(14,21)(15,42)(17,46)(18,51)(19,47)(20,48)(25,53)(26,55)(27,58)(28,56)(29,57)(32,59)(33,35)(34,52)(38,45)(39,63)(40,54)(43,62)(44,61)(49,60)(50,64)

Polycyclic presentation:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13311, #498817, #499270, …
maximal normal subgroup of	C₂×#13311, #498817, #499270, …
maximal quotient of	#142243, #142249, #142251, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6299);