#119397

order	512
nilpotency class	3
isoclinism	isoclinic class of #119392
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

531 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:

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

Polycyclic presentation:

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

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, 26593);