#6569663

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3307	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2043 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,e^c=ei,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6476859);