#263417

order	512
nilpotency class	3
isoclinism	isoclinic class of #263403
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

795 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 795 conjugacy classes of subgroups — too many to draw. Only its 144 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)(3,9)(4,12)(5,8)(6,10)(7,13)(11,16)(14,15)(17,18,23,28)(19,25,31,39)(20,32,35,46)(21,36,22,27)(24,37,40,30)(26,34,42,33)(29,43,45,38)(41,47,48,44);(4,13)(6,15)(7,12)(10,14)(17,19,22,37,23,31,21,30)(18,24,36,39,28,40,27,25)(20,29,34,47,35,45,33,44)(26,38,46,48,42,43,32,41);(1,3,5,11)(2,6,8,15)(4,14,13,10)(7,16,12,9)(17,19)(18,25)(20,29)(21,30)(22,37)(23,31)(24,36)(26,41)(27,40)(28,39)(32,43)(33,44)(34,47)(35,45)(38,46)(42,48);(17,20,22,34,23,35,21,33)(18,26,36,46,28,42,27,32)(19,29,37,47,31,45,30,44)(24,38,39,48,40,43,25,41)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,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, 170613);