#372562

order	512
nilpotency class	3
isoclinism	isoclinic class of #25443
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1210 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)(3,8)(4,6)(5,7)(9,10)(11,22)(12,26,29,18)(13,17)(14,30)(15,21)(16,20)(19,24)(23,34,38,37)(25,33)(27,40,39,32)(28,35,31,36);(1,3,5,4)(2,6,7,8)(9,11,14,13,15,24,16,25)(10,17,20,22,21,33,30,19)(12,27,31,23,29,39,28,38)(18,34,36,40,26,37,35,32);(1,4,5,3)(2,6,7,8)(9,12,14,28,15,29,16,31)(10,18,20,35,21,26,30,36)(11,23,13,27,24,38,25,39)(17,32,22,37,33,40,19,34);(2,7)(6,8)(9,13,14,24,15,25,16,11)(10,19,20,17,21,22,30,33)(12,27,28,38,29,39,31,23)(18,34,35,32,26,37,36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=e,d^2=fg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279758);