#115536

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, socle	C₂³	#392
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

802 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=h,e^2=h,f^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, 22732);