#119299

order	512
nilpotency class	3
isoclinism	isoclinic class of #119290
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

546 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)(3,7,11,14)(4,6)(5,9)(8,12)(10,13,16,15)(17,18,21,26,22,25,35,40)(19,27,29,41,30,23,34,37)(20,31,33,45,32,24,38,39)(28,36,44,47,43,42,46,48);(2,6)(3,10)(4,12)(7,14)(8,9)(11,16)(17,19,22,30)(18,23,25,27)(20,28,32,43)(21,34,35,29)(24,36,31,42)(26,41,40,37)(33,46,38,44)(39,48,45,47);(1,3,4,10,5,11,12,16)(2,7,8,15,9,14,6,13)(17,20)(18,24,25,31)(19,28)(21,33)(22,32)(23,36,27,42)(26,39,40,45)(29,44)(30,43)(34,46)(35,38)(37,47,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=d,d^2=i,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,e^c=eh,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26495);