#116565

order	512
nilpotency class	3
isoclinism	isoclinic class of #111253
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

704 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 170 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=di,d^2=h,e^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,e^b=eh,e^c=ehi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23761);