#236266

order	512
nilpotency class	3
isoclinism	isoclinic class of #236265
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	#60958	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1529 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:

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

Polycyclic presentation:

2^9|b^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,d^c=dh,e^c=ei,f^c=fhi

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, 143462);