#349989

order	512
nilpotency class	3
isoclinism	isoclinic class of #346398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1093 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=hi,c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,d^b=dh,f^c=fh,g^c=gi,f^d=fi

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, 257185);