#119376

order	512
nilpotency class	3
isoclinism	isoclinic class of #119374
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	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

515 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fhi,c^b=cf,d^b=di,e^b=ehi,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, 26572);