#390960

order	512
nilpotency class	3
isoclinism	isoclinic class of #232169
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

794 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=efg,c^2=egh,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 298156);