#329577

order	512
nilpotency class	3
isoclinism	isoclinic class of #329551
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#29662	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

722 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)(9,10,14,17,13,18,12,16)(11,19,20,23,21,15,22,24)(25,26,30,32,29,33,28,34)(27,31,37,39,36,40,35,38);(2,5)(6,8)(9,11)(10,15)(12,22)(13,21)(14,20)(16,23)(17,24)(18,19)(25,27)(26,31)(28,35)(29,36)(30,37)(32,39)(33,40)(34,38);(1,3)(2,6)(4,7)(5,8)(10,16)(12,14)(15,23)(17,18)(19,24)(20,22)(26,32)(27,35)(28,30)(33,34)(36,37)(38,39);(3,7)(6,8)(9,12,13,14)(10,17,18,16)(11,20,21,22)(15,23,19,24)(26,33)(27,36)(32,34)(35,37)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 236773);