#245384

order	512
nilpotency class	3
isoclinism	isoclinic class of #245375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1359 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1359 conjugacy classes of subgroups — too many to draw. Only its 178 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)(3,5)(4,7)(6,8)(9,10,14,19)(11,20,24,21)(12,26,28,34)(13,29,15,18)(16,25,32,23)(17,30,33,27)(22,31,38,40)(35,39,36,37);(9,11,14,24)(10,16,19,32)(12,22,28,38)(13,25,15,23)(17,31,33,40)(18,21,29,20)(26,35,34,36)(27,39,30,37);(2,5)(3,8)(6,7)(9,10)(11,21)(12,17)(13,18)(14,19)(15,29)(16,23)(20,24)(22,36)(25,32)(26,27)(28,33)(30,34)(31,39)(35,38)(37,40);(9,12,14,28)(10,17,19,33)(11,22,24,38)(13,30,15,27)(16,31,32,40)(18,34,29,26)(20,35,21,36)(23,39,25,37)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh

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, 152580);