#238992

order	512
nilpotency class	3
isoclinism	isoclinic class of #238990
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

874 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 182 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=ghi,d^c=dh,f^c=fhi,g^c=gi

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, 146188);