#9314

order	256
nilpotency class	3
isoclinism	isoclinic class of #9307
automorphism	Aut(G) = C₂³×D₄×#120 (order 2048) · Inn(G) = #386 · Out(G) = C₂⁵
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#386
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

203 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 203 conjugacy classes of subgroups — too many to draw. Only its 59 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 — 3 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#9314, #199268, #199269, …
maximal normal subgroup of	C₂×#9314, #199268, #199269, …
maximal quotient of	#99973, #99974, #99975, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 2302);