#243028

order	512
nilpotency class	3
isoclinism	isoclinic class of #243027
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
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

713 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,f^b=fh,d^c=dh,f^c=fi,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, 150224);