#112832

order	512
nilpotency class	3
isoclinism	isoclinic class of #100101
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 52 points:

(1,2)(3,4)(5,6)(7,14,16,11)(8,12)(9,13)(10,15,20,17)(18,19)(21,22)(23,33,37,28)(24,38)(25,32)(26,31)(27,30)(29,40)(34,46,51,52)(35,44,36,45)(39,43,50,49)(41,47)(42,48);(5,7)(6,10)(8,17)(9,16)(11,12)(13,20)(14,19)(15,18)(21,23,27,37)(22,28,30,33)(24,39,42,50)(25,36,26,35)(29,46,47,52)(31,45,32,44)(34,40,51,41)(38,49,48,43);(2,4)(5,7)(6,11)(8,17)(9,16)(10,19)(12,20)(13,14)(15,18)(21,24,27,42)(22,29,30,47)(23,34,37,51)(25,41,26,40)(28,43,33,49)(31,48,32,38)(35,39,36,50)(44,46,45,52)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fhi,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,d^c=di,e^c=eh,f^c=fh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 20028);