#391976

order	512
nilpotency class	3
isoclinism	isoclinic class of #232147
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

816 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 816 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 64

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

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

Polycyclic presentation:

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

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, 299172);