#122116

order	512
nilpotency class	3
isoclinism	isoclinic class of #122112
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1446
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

591 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fi,d^2=h,e^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,e^b=eh,d^c=di,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29312);