#120558

order	512
nilpotency class	3
isoclinism	isoclinic class of #120555
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

399 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27754);