#6544182

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2740 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)(21,22,27,39,26,30,25,31)(23,32,38,28,35,45,34,42)(24,29,40,50,36,44,37,43)(33,46,49,41,47,52,48,51);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,26,35)(22,28,30,42)(24,33,36,47)(25,38,27,34)(29,41,44,51)(31,45,39,32)(37,49,40,48)(43,52,50,46);(2,4)(6,10)(11,17)(12,18)(19,20)(21,24,26,36)(22,29,30,44)(23,33,35,47)(25,37,27,40)(28,41,42,51)(31,43,39,50)(32,46,45,52)(34,48,38,49);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(22,30)(24,36)(28,42)(31,39)(32,45)(33,47)(37,40)(48,49);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,d^2=h,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,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, 6451378);