#479886

order	512
nilpotency class	3
isoclinism	isoclinic class of #478213
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2842
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

618 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 170 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egh,c^2=g,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,e^c=ei,h^c=hi,f^d=fi,h^d=hi

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, 387082);