#329558

order	512
nilpotency class	3
isoclinism	isoclinic class of #329551
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

651 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)(9,10,14,18,13,17,22,24)(11,19,20,16,12,15,21,23)(25,26,28,33,29,32,37,31)(27,30,34,38,35,40,36,39);(2,5)(6,8)(9,11)(10,15)(12,13)(14,20)(16,24)(17,19)(18,23)(21,22)(25,27,29,35)(26,30,32,40)(28,34,37,36)(31,39,33,38);(1,3)(2,6)(4,7)(5,8)(9,12,13,11)(10,16,17,23)(14,20,22,21)(15,18,19,24)(26,31)(27,34)(28,37)(32,33)(35,36)(38,39);(3,7)(6,8)(10,17)(11,12)(18,24)(20,21)(26,32)(27,35)(31,33)(34,36)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^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, 236754);