#390964

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

807 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)(9,10,14,21)(11,16)(12,24,29,37)(13,30)(15,20,32,19)(17,28,36,27)(18,31)(22,35)(23,34)(25,39)(26,40)(33,38);(1,3,5,2)(4,7,8,6)(9,11)(10,16)(12,25)(13,14)(15,31)(17,35)(18,20)(19,34)(21,30)(22,28)(23,32)(24,33)(26,27)(29,38)(36,40)(37,39);(1,4,5,8)(2,6,3,7)(9,12,15,27,14,29,32,28)(10,17,19,24,21,36,20,37)(11,22,31,25,13,26,23,38)(16,33,34,40,30,39,18,35);(2,3)(6,7)(9,13,14,11)(10,18,21,34)(12,26,29,22)(15,23,32,31)(16,20,30,19)(17,35,36,40)(24,33,37,39)(25,27,38,28)

Polycyclic presentation:

2^9|a^2=h,b^2=efg,c^2=egh,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,f^d=fi,g^d=gi,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, 298160);