#12832

order	256
nilpotency class	4
isoclinism	isoclinic class of #12026
automorphism	Aut(G) = #72313012 (order 4096) · Inn(G) = #427 · Out(G) = C₂×C₄×D₄
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#427
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1807
prime radical	C₂×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

179 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 179 conjugacy classes of subgroups — too many to draw. Only its 77 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)(3,13,16,34)(4,18,20,9)(5,22)(6,25)(7,11)(8,15,28,40)(10,23)(12,26)(14,36)(17,43,42,35)(19,47,48,51)(21,31,50,46)(24,52)(27,37)(29,56,54,41)(30,49,45,63)(32,53)(33,39)(38,57)(44,61)(55,64)(58,60)(59,62);(1,3,5,15,6,17,23,41,7,16,24,40,26,42,53,56)(2,8,10,13,12,29,22,35,11,28,32,34,25,54,52,43)(4,14,19,37,21,38,49,60,20,39,48,61,50,62,63,64)(9,27,30,33,31,55,51,57,18,44,45,36,46,58,47,59);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,27)(10,30)(11,18)(12,31)(13,33)(15,37)(16,39)(17,38)(22,51)(23,49)(24,48)(25,46)(26,50)(28,44)(29,55)(32,45)(34,36)(35,57)(40,61)(41,60)(42,62)(43,59)(47,52)(53,63)(54,58)(56,64)

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#134747, #134748, #134749, …
maximal normal subgroup of	#134747, #134748, #134749, …
maximal quotient of	#134659, #134660, #134691, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5820);