#119780

order	512
nilpotency class	3
isoclinism	isoclinic class of #119775
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

529 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=h,e^2=h,f^2=i,g^2=h,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=fi,f^c=fhi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26976);