#349822

order	512
nilpotency class	3
isoclinism	isoclinic class of #346294
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	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1182 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)(9,10,13,17)(11,14,19,23)(12,15,20,16)(18,21,24,22)(33,39)(36,40);(2,5)(6,8)(9,11,13,19)(10,14,17,23)(12,18,20,24)(15,21,16,22)(25,26,29,32)(27,30,34,37)(28,31,35,38)(33,36,39,40);(1,3,4,7)(2,6,5,8)(10,15)(12,20)(14,21)(16,17)(18,24)(22,23)(25,27)(26,30)(28,33)(29,34)(31,36)(32,37)(35,39)(38,40);(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=e,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,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, 257018);