#351513

order	512
nilpotency class	3
isoclinism	isoclinic class of #348398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

602 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,22)(19,21)(23,24)(25,26,30,33)(27,34,37,40)(28,32,38,39)(29,36,31,35);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,21)(17,23)(20,24)(25,27,31,38,30,37,29,28)(26,32,36,40,33,39,35,34);(2,6)(5,8)(10,15)(11,18)(16,17)(19,24)(26,33)(27,37)(28,38)(35,36);(10,16)(13,20)(14,21)(15,17)(19,24)(22,23)(25,28,29,37,30,38,31,27)(26,34,35,39,33,40,36,32)

Polycyclic presentation:

2^9|a^2=h,b^2=efhi,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=ci,d^c=dh,g^d=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 258709);