#329514

order	512
nilpotency class	3
isoclinism	isoclinic class of #329455
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

681 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)(3,6)(4,5)(7,8)(9,10,15,23,14,19,13,18)(11,16,20,24,12,22,21,17)(25,26)(27,35)(28,33)(29,32)(30,34)(31,38)(36,39)(37,40);(2,5)(6,8)(9,11)(10,16)(12,14)(13,21)(15,20)(17,18)(19,22)(23,24)(25,27)(26,31)(28,37)(29,38)(30,36)(32,35)(33,39)(34,40);(1,3)(2,6)(4,7)(5,8)(9,12,14,11)(10,17,19,24)(13,21,15,20)(16,23,22,18)(26,32)(27,36)(28,30)(31,39)(35,40)(37,38);(3,7)(6,8)(9,13,14,15)(10,18,19,23)(11,20,12,21)(16,24,22,17)(25,28,29,30)(26,33,32,34)(27,37,38,36)(31,39,35,40)

Polycyclic presentation:

2^9|a^2=fhi,c^2=h,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,c^b=cg,d^b=dh,g^b=gi,d^c=de,f^c=fh,g^c=gi

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, 236710);