#168104

order	512
nilpotency class	3
isoclinism	isoclinic class of #167761
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	#37216	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

785 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 785 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=ghi,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fi

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, 75300);