#403673

order	512
nilpotency class	3
isoclinism	isoclinic class of #395187
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 932 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=g,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^c=fi,h^c=hi,f^d=fi

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, 310869);