#118905

order	512
nilpotency class	3
isoclinism	isoclinic class of #118896
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	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

559 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)(3,11,14,16)(4,7)(5,9)(6,10)(8,13,15,12)(17,18,22,27,23,26,21,38)(19,28,31,44,32,24,30,40)(20,33,36,42,37,43,35,25)(29,41,34,39,47,48,46,45);(2,7)(3,12)(4,5)(8,15)(9,10)(13,14)(17,19,23,32)(18,24,26,28)(20,34,37,46)(21,31,22,30)(25,41,42,48)(27,44,38,40)(29,35,47,36)(33,45,43,39);(1,3,4,13,6,14,5,12)(2,8,9,16,10,15,7,11)(17,20)(18,25,26,42)(19,29)(21,35)(22,36)(23,37)(24,39,28,45)(27,33,38,43)(30,46)(31,34)(32,47)(40,48,44,41)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=d,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,f^c=fhi,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, 26101);