#13405

order	256
nilpotency class	4
isoclinism	isoclinic class of #13404
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#2034
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

193 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 193 conjugacy classes of subgroups — too many to draw. Only its 53 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13405, #499043, #499604, …
maximal normal subgroup of	C₂×#13405, #499043, #499604, …
maximal quotient of	#144719, #144747, #145028, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6393);