#411613

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1834 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)(3,4)(9,10,15,20)(11,21,24,36)(12,25,29,34)(13,19,14,18)(16,30,32,23)(17,27,35,28)(22,33,26,31)(37,40,39,38);(2,4)(5,6)(7,8)(9,11)(10,16)(12,26)(13,30)(14,23)(15,24)(17,33)(18,36)(19,21)(20,32)(22,29)(25,39)(27,38)(28,40)(31,35)(34,37);(6,8)(9,11)(10,16)(12,22)(13,30)(14,23)(15,24)(17,31)(18,36)(19,21)(20,32)(25,37)(26,29)(27,40)(28,38)(33,35)(34,39);(9,12,13,27,15,29,14,28)(10,17,18,34,20,35,19,25)(11,22,30,40,24,26,23,38)(16,31,36,39,32,33,21,37)

Polycyclic presentation:

2^9|a^2=i,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,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, 318809);