#239976

order	512
nilpotency class	3
isoclinism	isoclinic class of #239961
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

907 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)(3,7)(4,6)(5,8)(9,10,15,19)(11,20,24,35)(12,25,29,17)(13,18,14,30)(16,22,32,23)(21,37,26,36)(27,40,28,34)(31,39,33,38);(9,11,15,24)(10,16,19,32)(12,26,29,21)(13,23,14,22)(17,33,25,31)(18,35,30,20)(27,38,28,39)(34,36,40,37);(2,6)(3,4)(7,8)(9,12,14,28,15,29,13,27)(10,17,18,34,19,25,30,40)(11,21,23,39,24,26,22,38)(16,31,20,36,32,33,35,37);(1,3,5,4)(2,7,8,6)(9,13,15,14)(10,18,19,30)(11,22,24,23)(12,27,29,28)(16,20,32,35)(17,34,25,40)(21,38,26,39)(31,36,33,37)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,d^b=dh,e^b=eh,d^c=di,f^c=fhi,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, 147172);