#215139

order	512
nilpotency class	3
isoclinism	isoclinic class of #213537
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

950 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 950 conjugacy classes of subgroups — too many to draw. Only its 226 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)(11,17,18,15)(12,20)(13,14)(16,21)(19,22,24,23)(25,26,28,32,29,31,35,38)(27,30,34,37,33,36,39,40);(1,3,4,8)(2,5,6,7)(10,14)(11,18)(16,20)(19,24)(27,33)(30,36)(34,39)(37,40);(3,8)(5,7)(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,27)(26,30)(28,34)(29,33)(31,36)(32,37)(35,39)(38,40);(2,6)(5,7)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(26,31)(27,33)(32,38)(34,39)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=h,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cei,d^c=di,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, 122335);