#326593

order	512
nilpotency class	3
isoclinism	isoclinic class of #251818
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2580
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

413 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=ei,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,f^b=fh,d^c=dh,f^c=fhi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233789);