#115689

order	512
nilpotency class	3
isoclinism	isoclinic class of #115666
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
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

764 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=dh,f^b=fhi,d^c=dh,f^c=fhi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22885);