#240070

order	512
nilpotency class	3
isoclinism	isoclinic class of #240066
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

921 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:

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

Polycyclic presentation:

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

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, 147266);