#6569728

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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×#3321	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2042 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)(28,35)(33,46)(34,42)(36,47)(37,43)(38,44)(39,45)(41,48)(49,52)(50,51);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,35,27,36,25,34)(22,28,30,32,31,42,40,47)(24,33,37,48,39,50,38,49)(29,41,44,46,45,52,43,51);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(21,24)(22,29)(23,33)(25,38)(26,37)(27,39)(28,41)(30,44)(31,45)(32,46)(34,49)(35,48)(36,50)(40,43)(42,52)(47,51);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(21,25,27,26)(22,30,31,40)(23,34,36,35)(24,37,39,38)(28,32,42,47)(29,43,45,44)(33,48,50,49)(41,51,52,46);(8,15)(11,18)(14,19)(17,20)(21,25,27,26)(22,30,31,40)(23,34,36,35)(24,38,39,37)(28,32,42,47)(29,44,45,43)(33,49,50,48)(41,46,52,51)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,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, 6476924);