#326587

order	512
nilpotency class	3
isoclinism	isoclinic class of #251809
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

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

normal subgroups only

This group has 423 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:

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

Polycyclic presentation:

2^9|a^2=i,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,f^c=fhi

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, 233783);