#331027

order	512
nilpotency class	3
isoclinism	isoclinic class of #331007
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
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

516 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ei,c^2=h,d^2=fhi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,g^b=gi,d^c=de,e^c=eh,f^c=fh,g^c=gi,g^d=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238223);