#13403

order	256
nilpotency class	4
isoclinism	isoclinic class of #13402
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

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

normal subgroups only

This group has 192 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)(3,13,18,33)(4,19,24,54)(5,25,27,10)(6,28,30,52)(8,17,34,15)(9,35,39,22)(11,40,42,26)(14,44,47,64)(16,32,51,62)(20,55,45,63)(21,53,23,41)(29,38,56,37)(31,59,61,49)(36,58,60,46)(43,48,57,50);(1,3)(2,8)(4,20)(5,17)(6,16)(7,18)(9,36)(10,33)(11,32)(12,34)(13,25)(14,23)(15,27)(19,44)(21,47)(22,49)(24,45)(26,50)(28,57)(29,58)(30,51)(31,38)(35,59)(37,61)(39,60)(40,48)(41,63)(42,62)(43,52)(46,56)(53,55)(54,64);(1,4,7,24)(2,9,12,39)(3,14)(5,23,27,21)(6,29,30,56)(8,31)(10,38,25,37)(11,41,42,53)(13,36)(15,20)(16,49)(17,45)(18,47)(19,52,54,28)(22,40,35,26)(32,44)(33,60)(34,61)(43,63)(46,50)(48,58)(51,59)(55,57)(62,64)

Polycyclic presentation:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13403, #499040, #499515, …
maximal normal subgroup of	C₂×#13403, #499040, #499515, …
maximal quotient of	#144721, #144749, #145028, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6391);