#120333

order	512
nilpotency class	3
isoclinism	isoclinic class of #120331
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1441
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

404 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:

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

Polycyclic presentation:

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

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, 27529);