#412153

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1550 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=e,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=di,f^c=fi

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, 319349);