#345863

order	512
nilpotency class	3
isoclinism	isoclinic class of #345848
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	C₂×C₄²	C₂⁴
prime radical	#3497	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 594 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)(11,21,26,23)(12,27,32,36)(14,20,15,19)(17,25,33,24)(18,31,29,30)(22,37,40,34)(28,38,39,35);(1,2,4,7)(3,5,8,6)(9,11,15,25,16,26,14,24)(10,17,20,23,13,33,19,21)(12,28,30,22,32,39,31,40)(18,35,27,37,29,38,36,34);(2,5)(3,8)(6,7)(9,12)(10,18)(11,22)(13,29)(14,30)(15,31)(16,32)(17,34)(19,27)(20,36)(21,35)(23,38)(24,39)(25,28)(26,40)(33,37);(9,13,16,10)(11,23,26,21)(12,29,32,18)(14,19,15,20)(17,24,33,25)(22,38,40,35)(27,31,36,30)(28,34,39,37)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,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=deh,g^b=gi,f^c=fh,g^c=ghi,e^d=eh,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253059);