#315890

order	512
nilpotency class	3
isoclinism	isoclinic class of #298867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

674 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 178 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ehi,c^2=f,d^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh,d^c=di

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, 223086);