#326523

order	512
nilpotency class	3
isoclinism	isoclinic class of #326513
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2563
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#535
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

414 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=egi,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,f^a=fh,c^b=cg,f^c=fi,g^c=gh

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, 233719);