#351123

order	512
nilpotency class	3
isoclinism	isoclinic class of #347788
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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

598 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)(10,14)(11,17)(12,19)(15,22)(16,21)(18,23)(25,26)(27,36)(28,38)(29,33)(30,35)(31,32)(34,37)(39,40);(2,5)(6,8)(9,10,12,14,13,16,19,21)(11,15,17,20,18,22,23,24)(25,27,30,31)(26,32,35,36)(28,39,29,37)(33,40,38,34);(1,3,4,7)(2,6,5,8)(9,11,12,17,13,18,19,23)(10,15,14,20,16,22,21,24)(25,28,30,29)(26,33,35,38)(27,37,31,39)(32,34,36,40);(10,16)(14,21)(15,22)(20,24)(25,29,30,28)(26,34,35,40)(27,37,31,39)(32,33,36,38)

Polycyclic presentation:

2^9|b^2=f,c^2=e,d^2=i,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=gi,c^b=ci,d^b=dh,g^d=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, 258319);