#121107

order	512
nilpotency class	3
isoclinism	isoclinic class of #118505
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#388
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1443
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

463 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28303);