#119871

order	512
nilpotency class	3
isoclinism	isoclinic class of #119867
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

506 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27067);