#387204

order	512
nilpotency class	3
isoclinism	isoclinic class of #232128
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

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

678 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efg,c^2=g,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,h^c=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, 294400);