#13335

order	256
nilpotency class	4
isoclinism	isoclinic class of #13334
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	#2011
commutator, Frattini, cosocle	C₂²×C₈	C₂³
prime radical	#3414	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

268 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 59 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)(4,18)(5,23)(6,29)(7,11)(8,15)(9,20)(10,27)(12,26)(14,32)(16,54)(17,46)(19,37)(21,59)(22,36)(24,42)(25,43)(28,44)(30,39)(31,40)(33,53)(34,60)(35,50)(38,58)(41,51)(45,48)(47,56)(49,64)(52,62)(55,63)(57,61);(1,3,7,17)(2,8,11,33)(4,14,22,50)(5,24,28,60)(6,30,26,16)(9,35,37,32)(10,39,41,54)(12,42,29,34)(13,44,46,23)(15,51,53,27)(18,56,36,45)(19,57,20,52)(21,55,31,49)(25,48,58,47)(38,63,43,64)(40,61,59,62);(1,4)(2,9)(3,14)(5,25)(6,31)(7,22)(8,32)(10,40)(11,37)(12,43)(13,45)(15,52)(16,55)(17,50)(18,23)(19,51)(20,27)(21,26)(24,48)(28,58)(29,38)(30,49)(33,35)(34,64)(36,44)(39,62)(41,59)(42,63)(46,56)(47,60)(53,57)(54,61)

Polycyclic presentation:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13335, #498872, #499270, …
maximal normal subgroup of	C₂×#13335, #498872, #499270, …
maximal quotient of	#142626, #142627, #142630, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6323);