#6578467

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2057 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,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, 6485663);