#349893

order	512
nilpotency class	3
isoclinism	isoclinic class of #346346
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1152 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)(4,5)(7,8)(9,10,14,17)(11,15,13,22)(12,19,21,23)(16,20,24,18)(25,26,29,32)(27,30,34,37)(28,31,35,38)(33,39,40,36);(2,5)(6,8)(9,11,14,13)(10,15,17,22)(12,20,21,18)(16,19,24,23);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,18)(13,20)(14,21)(15,23)(17,24)(19,22)(25,27)(26,30)(28,33)(29,34)(31,36)(32,37)(35,40)(38,39);(9,11,14,13)(10,15,17,22)(12,18,21,20)(16,23,24,19)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,40)(37,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,c^b=ch,f^c=fh,g^c=gi,f^d=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, 257089);