#6544249

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2751 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)(21,22,27,40,26,30,25,31)(23,32,39,28,35,45,34,43)(24,36,41,44,37,29,38,51)(33,47,52,50,48,46,49,42);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,26,35)(22,28,30,43)(24,33,37,48)(25,39,27,34)(29,42,36,50)(31,45,40,32)(38,52,41,49)(44,47,51,46);(2,4)(6,10)(11,17)(12,18)(19,20)(21,24,26,37)(22,29,30,36)(23,33,35,48)(25,38,27,41)(28,42,43,50)(31,44,40,51)(32,46,45,47)(34,49,39,52);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(24,37)(29,36)(33,48)(38,41)(42,50)(44,51)(46,47)(49,52);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(22,30)(28,43)(29,36)(31,40)(32,45)(42,50)(44,51)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6451445);