#13500

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	#2076
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

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

normal subgroups only

This group has 138 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:

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

Polycyclic presentation:

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

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13500, #499400, #499595, …
maximal normal subgroup of	C₂×#13500, #499400, #499595, …
maximal quotient of	#146024, #146025, #146034, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6488);