#10125

order	256
nilpotency class	3
isoclinism	isoclinic class of #10120
automorphism	Aut(G) = C₂×#72278177 (order 4096) · Inn(G) = #392 · Out(G) = C₂×#120
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂²×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

229 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 229 conjugacy classes of subgroups — too many to draw. Only its 49 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 — 3 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#10125, #236604, #236611, …
maximal normal subgroup of	C₂×#10125, #236604, #236611, …
maximal quotient of	#102509, #102513, #102516, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 3113);