#386530

order	512
nilpotency class	3
isoclinism	isoclinic class of #219696
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

967 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=h,c^2=efg,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^b=ei,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, 293726);