#239091

order	512
nilpotency class	3
isoclinism	isoclinic class of #233220
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

904 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,19,23)(15,24,27,36)(16,28,32,21)(17,22,18,33)(20,34,37,26)(25,41,29,40)(30,43,31,39)(35,42,38,44);(5,6)(7,11)(8,10)(9,12)(13,15,19,27)(14,20,23,37)(16,29,32,25)(17,26,18,34)(21,38,28,35)(22,36,33,24)(30,44,31,42)(39,40,43,41);(2,4)(6,10)(7,8)(11,12)(13,16,18,31,19,32,17,30)(14,21,22,39,23,28,33,43)(15,25,26,42,27,29,34,44)(20,35,24,40,37,38,36,41);(5,7,9,8)(6,11,12,10)(13,17,19,18)(14,22,23,33)(15,26,27,34)(16,30,32,31)(20,36,37,24)(21,39,28,43)(25,42,29,44)(35,41,38,40)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gi,d^c=di,f^c=fhi,g^c=ghi

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, 146287);