#437011

order	512
nilpotency class	3
isoclinism	isoclinic class of #418863
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

766 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 766 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 64

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

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

Polycyclic presentation:

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

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, 344207);