#350010

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

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

1055 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)(11,18)(12,16)(13,15)(14,19)(17,21)(20,23)(22,24)(25,26,30,27)(28,31,36,33)(29,32,37,34)(35,39,40,38);(2,5)(6,8)(9,11)(10,14)(12,19)(13,20)(15,22)(16,18)(17,23)(21,24)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,39,40,38);(1,3,4,7)(2,6,5,8)(10,15)(13,21)(14,22)(16,17)(18,23)(20,24)(25,28)(26,31)(27,33)(29,35)(30,36)(32,38)(34,39)(37,40);(11,19)(14,18)(20,24)(22,23)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=eh,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, 257206);