#12827

order	256
nilpotency class	4
isoclinism	isoclinic class of #12826
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#1804
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
prime radical	C₂³	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

188 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 69 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)(3,14,18,33)(4,19)(5,10)(6,26)(7,13)(8,16,32,49)(9,21)(11,24)(12,25)(15,43,46,64)(17,31,27,38)(20,52)(22,53)(23,39)(28,34)(29,37)(30,59,62,50)(35,57)(36,56)(40,55)(41,54,60,45)(42,58,63,48)(44,61,47,51);(1,3,7,18)(2,8)(4,15,22,46)(5,16,25,49)(6,27,29,17)(9,30)(10,14)(11,38)(12,33)(13,32)(19,51)(20,44,23,47)(21,54,56,45)(24,58,55,48)(26,42)(28,59,57,50)(31,40)(34,60)(35,41)(36,62)(37,63)(39,64)(43,52)(53,61);(1,4,7,22)(2,9,13,36)(3,15,18,46)(5,23,25,20)(6,28,29,57)(8,30,32,62)(10,35,12,34)(11,39,40,52)(14,41,33,60)(16,47,49,44)(17,50,27,59)(19,37,53,26)(21,55,56,24)(31,43,38,64)(42,51,63,61)(45,58,54,48)

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#12827, #493794, #494059, …
maximal normal subgroup of	C₂×#12827, #493794, #494059, …
maximal quotient of	#134391, #134416, #134476, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5815);