#119821

order	512
nilpotency class	3
isoclinism	isoclinic class of #119819
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

537 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,9,12,6)(4,13)(5,7)(8,15)(10,14,16,11)(17,18,22,28,23,26,21,27)(19,29,32,39,33,24,31,40)(20,34,35,42,36,43,37,25)(30,44,45,38,46,48,47,41);(1,2)(3,10)(4,13)(5,7)(6,11)(8,15)(9,14)(12,16)(17,19,23,33)(18,24,26,29)(20,30,36,46)(21,32,22,31)(25,41,42,38)(27,40,28,39)(34,48,43,44)(35,47,37,45);(1,3)(2,6)(4,14)(5,12)(7,9)(8,16)(10,13)(11,15)(17,20)(18,25,26,42)(19,30)(21,35)(22,37)(23,36)(24,38,29,41)(27,34,28,43)(31,45)(32,47)(33,46)(39,48,40,44)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,e^b=ehi,f^b=fi,g^b=gh,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27017);