#118768

order	512
nilpotency class	3
isoclinism	isoclinic class of #118765
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	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

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

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, 25964);