#351477

order	512
nilpotency class	3
isoclinism	isoclinic class of #348438
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

571 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)(11,18)(12,15)(13,16)(14,19)(17,21)(20,23)(22,24)(25,26,31,34)(27,35,36,37)(28,30,33,29)(32,39,40,38);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,19)(13,20)(15,18)(16,22)(17,23)(21,24)(25,27,30,39,31,36,29,38)(26,32,28,37,34,40,33,35);(2,6)(5,8)(10,15)(11,19)(16,17)(20,24)(27,36)(32,40)(35,37)(38,39);(10,16)(13,21)(14,22)(15,17)(18,23)(20,24)(25,28,30,34,31,33,29,26)(27,37,39,40,36,35,38,32)

Polycyclic presentation:

2^9|a^2=h,b^2=efh,d^2=ei,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=ci,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, 258673);