#303872

order	512
nilpotency class	3
isoclinism	isoclinic class of #268139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2819	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

809 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 809 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=efh,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,g^c=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211068);