#6406443

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1273 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fg,d^2=g,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=efi,c^b=ci,d^b=dfi,e^b=ei,e^d=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6313639);