#345825

order	512
nilpotency class	3
isoclinism	isoclinic class of #345808
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2684
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#534
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

501 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 120 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=def,e^a=eh,f^a=fh,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=ghi,e^d=eh,f^d=fh

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, 253021);