#386528

order	512
nilpotency class	3
isoclinism	isoclinic class of #219532
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

982 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)(4,7)(6,8)(9,10,16,21,15,22,14,20)(11,23,13,17,26,34,25,19)(12,27,30,36,31,37,32,18)(24,35,39,40,28,33,29,38);(1,3)(2,4)(5,8)(6,7)(9,11,15,26)(10,17,22,19)(12,28,31,24)(13,16,25,14)(18,33,36,35)(20,34,21,23)(27,40,37,38)(29,30,39,32);(1,4,6,5)(2,7,8,3)(9,12)(10,18,22,36)(11,24)(13,29)(14,30)(15,31)(16,32)(17,33,19,35)(20,27,21,37)(23,38,34,40)(25,39)(26,28);(1,5,6,4)(2,7,8,3)(9,13,14,11,15,25,16,26)(10,19,20,34,22,17,21,23)(12,29,30,24,31,39,32,28)(18,35,27,40,36,33,37,38)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=efg,d^2=e,e^2=i,f^2=i,h^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,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, 293724);