#18472

order	256
nilpotency class	3
isoclinism	isoclinic class of #18446
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	C₂²×C₄	C₂⁴
prime radical	#538	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

423 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 423 conjugacy classes of subgroups — too many to draw. Only its 111 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#347420, #347421, #347424, …
maximal normal subgroup of	#347420, #347421, #347424, …
maximal quotient of	#168944, #168960, #168999, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 11460);