#115541

order	512
nilpotency class	3
isoclinism	isoclinic class of #10120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

801 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=h,e^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=dh,f^b=fhi,d^c=di,e^c=eh,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, 22737);