#6571718

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#38032	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2824 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6478914);