#382704

order	512
nilpotency class	3
isoclinism	isoclinic class of #213736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1741 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1741 conjugacy classes of subgroups — too many to draw. Only its 274 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:

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

Polycyclic presentation:

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

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, 289900);