#478422

order	512
nilpotency class	3
isoclinism	isoclinic class of #476654
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

976 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 154 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)(6,8)(9,10,14,16,15,20,13,19)(11,21,24,33,25,31,23,17)(12,26,28,36,30,35,29,18)(22,37,27,40,39,32,38,34);(1,2,4,5)(3,6,7,8)(10,16)(13,14)(17,31)(18,26)(19,20)(21,33)(23,24)(27,38)(28,29)(32,40)(34,37)(35,36);(1,3)(2,6)(4,7)(5,8)(9,11,15,25)(10,17,20,33)(12,27,30,38)(13,24,14,23)(16,31,19,21)(18,34,36,40)(22,29,39,28)(26,32,35,37);(2,5)(3,7)(9,12)(10,18)(11,22)(13,29)(14,28)(15,30)(16,26)(17,32)(19,35)(20,36)(21,34)(23,38)(24,27)(25,39)(31,40)(33,37)

Polycyclic presentation:

2^9|a^2=fg,b^2=ef,c^2=i,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=def,e^b=ei,f^b=fi,h^b=hi,d^c=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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385618);