#13035

order	256
nilpotency class	4
isoclinism	isoclinic class of #13033
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#1879
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

228 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 67 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:

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

Polycyclic presentation:

2^8|a^2=f,b^2=dfg,d^2=g,e^2=g,f^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=dh,d^b=dg,e^b=egh,f^b=fg,g^b=gh,d^c=dgh,e^c=egh,f^c=fgh,g^c=gh

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#13035, #494556, #495738, …
maximal normal subgroup of	C₂×#13035, #494556, #495738, …
maximal quotient of	#136701, #136702, #136735, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 6023);