#121291

order	512
nilpotency class	3
isoclinism	isoclinic class of #121286
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=dh,d^2=i,e^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=ghi,e^c=ehi,f^c=fhi

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, 28487);