#339864

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=ei,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, 247060);