#6510843

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500416
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	#37455	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

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,e^b=eh,d^c=dh,e^c=ei

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, 6418039);