#119276

order	512
nilpotency class	3
isoclinism	isoclinic class of #119271
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	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

552 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,f^b=fi,g^b=ghi,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, 26472);