#452978

order	512
nilpotency class	3
isoclinism	isoclinic class of #417903
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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 950 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,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, 360174);