#236531

order	512
nilpotency class	3
isoclinism	isoclinic class of #236515
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60907	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1246 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)(11,16)(12,24)(13,18)(14,27)(15,19)(17,25)(20,35)(21,30)(22,34)(23,31)(26,32)(28,33)(29,36)(37,39)(38,40);(2,5)(3,8)(6,7)(9,11,15,23)(10,16,19,31)(12,20,26,37)(13,22,14,21)(17,29,33,40)(18,34,27,30)(24,39,32,35)(25,38,28,36);(1,2)(3,6)(4,7)(5,8)(9,12)(10,17)(11,20)(13,25)(14,28)(15,26)(16,29)(18,32)(19,33)(21,36)(22,38)(23,37)(24,27)(30,39)(31,40)(34,35);(9,13,15,14)(10,18,19,27)(11,21,23,22)(12,25,26,28)(16,30,31,34)(17,32,33,24)(20,36,37,38)(29,39,40,35)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 143727);