#256416

order	512
nilpotency class	3
isoclinism	isoclinic class of #248959
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2656	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

854 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 854 conjugacy classes of subgroups — too many to draw. Only its 142 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)(3,8)(4,6)(5,7)(9,10)(11,16)(12,21)(13,18)(14,23)(15,19)(17,20)(22,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,36)(35,38)(39,40);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,14)(13,22)(15,20)(17,19)(18,24)(21,23)(25,27)(26,30)(28,33)(29,34)(31,37)(32,36)(35,39)(38,40);(1,4,5,3)(2,6,7,8)(9,12,15,22)(10,17,19,16)(11,13,20,14)(18,24,23,21)(30,36)(37,40);(2,7)(6,8)(9,13,15,14)(10,18,19,23)(11,12,20,22)(16,21,17,24)(25,28,29,35)(26,31,32,38)(27,33,34,39)(30,37,36,40)

Polycyclic presentation:

2^9|b^2=e,c^2=eh,d^2=hi,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,f^c=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, 163612);