#339865

order	512
nilpotency class	3
isoclinism	isoclinic class of #339847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
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

511 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 511 conjugacy classes of subgroups — too many to draw. Only its 122 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)(3,11)(4,7)(5,9)(6,10)(8,12)(13,15)(14,16)(17,18,24,20)(19,28,32,43)(21,36,40,48)(22,35,23,27)(25,31,33,30)(26,39,34,38)(29,44,46,41)(37,42,45,47);(1,2)(3,8)(4,9)(5,7)(6,10)(11,12)(13,16)(14,15)(17,19,24,32)(18,25,20,33)(21,37,40,45)(22,31,23,30)(26,42,34,47)(27,43,35,28)(29,39,46,38)(36,41,48,44);(2,7)(4,5)(8,11)(9,10)(12,14)(15,16)(17,20,24,18)(19,28,32,43)(21,34,40,26)(22,35,23,27)(25,30,33,31)(29,42,46,47)(36,38,48,39)(37,44,45,41);(1,3,6,13)(2,8,10,16)(4,14,5,12)(7,11,9,15)(17,21,23,39,24,40,22,38)(18,26,35,48,20,34,27,36)(19,29,31,45,32,46,30,37)(25,41,43,47,33,44,28,42)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=ehi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 247061);