#19097

order	256
nilpotency class	3
isoclinism	isoclinic class of #19086
automorphism	Aut(G) = C₂×#72286737 (order 4096) · Inn(G) = C₂×#120 · Out(G) = C₂×#120
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3567	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

415 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 109 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(3,10)(4,5)(7,14)(8,12)(11,13)(15,16)(17,18,23,28)(19,29,31,41)(20,33,35,42)(21,27,22,26)(24,36,39,32)(25,37,40,34)(30,38,44,48)(43,45,47,46);(1,2)(3,7)(4,8)(5,12)(6,9)(10,15)(11,16)(13,14)(17,19,23,31)(18,24,28,39)(20,30,35,44)(21,36,22,32)(25,38,40,48)(26,41,27,29)(33,47,42,43)(34,46,37,45);(1,3)(2,7)(4,10)(5,13)(6,11)(8,15)(9,16)(12,14)(17,20)(18,25)(19,30)(21,34)(22,37)(23,35)(24,38)(26,33)(27,42)(28,40)(29,43)(31,44)(32,45)(36,46)(39,48)(41,47);(1,4,6,5)(2,8,9,12)(3,10,11,13)(7,15,16,14)(19,31)(24,39)(29,41)(30,44)(32,36)(38,48)(43,47)(45,46)

Polycyclic presentation:

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

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#347419, #347421, #347423, …
maximal normal subgroup of	#347419, #347421, #347423, …
maximal quotient of	#168936, #168943, #168944, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12085);