#237967

order	512
nilpotency class	3
isoclinism	isoclinic class of #237964
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 880 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,24,26,35)(13,18)(14,29)(15,19)(17,27,34,28)(20,33,36,40)(21,31)(22,32)(23,38)(25,30,37,39);(9,11,15,21)(10,16,19,31)(12,25)(13,22,14,23)(17,33)(18,32,29,38)(20,27)(24,39)(26,37)(28,36)(30,35)(34,40);(2,6)(3,4)(7,8)(9,12)(10,17)(11,20)(13,28)(14,27)(15,26)(16,30)(18,35)(19,34)(21,36)(22,25)(23,37)(24,29)(31,39)(32,40)(33,38);(1,3,5,4)(2,7,8,6)(11,21)(16,31)(20,36)(22,23)(25,37)(30,39)(32,38)(33,40)

Polycyclic presentation:

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

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, 145163);