#137258

order	512
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1874
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

337 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 337 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgi,d^2=gi,e^2=i,f^2=gh,g^2=i,b^a=bd,c^a=ce,e^a=ehi,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=ei,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44454);