#13435

order	256
nilpotency class	4
isoclinism	isoclinic class of #13432
automorphism	Aut(G) = C₂×#72236702 (order 2048) · Inn(G) = #2044 · Out(G) = C₂⁴
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#2044
commutator, Frattini, prime radical, cosocle	C₂×#41	C₂³
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

204 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 43 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)(3,13,17,35)(4,18,22,47)(5,23,26,10)(6,27,28,11)(8,24,31,15)(9,32,34,21)(14,39,19,48)(16,43,44,38)(20,51,53,46)(25,36,57,55)(29,59,61,64)(30,42,58,56)(33,52,54,62)(37,50,63,41)(40,60,49,45);(1,3,7,17)(2,8,12,31)(4,19,22,14)(5,24,26,15)(6,16,28,44)(9,29,34,61)(10,35,23,13)(11,30,27,58)(18,45,47,60)(20,40,53,49)(21,41,32,50)(25,56,57,42)(33,63,54,37)(36,43,55,38)(39,51,48,46)(52,59,62,64);(1,4,5,20,7,22,26,53)(2,9,10,33,12,34,23,54)(3,14,15,40,17,19,24,49)(6,21,25,52,28,32,57,62)(8,29,13,37,31,61,35,63)(11,18,36,46,27,47,55,51)(16,41,42,64,44,50,56,59)(30,60,38,39,58,45,43,48)

Polycyclic presentation:

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

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13435, #499150, #499152, …
maximal normal subgroup of	C₂×#13435, #499150, #499152, …
maximal quotient of	#144673, #144677, #144709, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6423);