#241396

order	512
nilpotency class	3
isoclinism	isoclinic class of #234079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

784 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148592);