#6578194

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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	#32911	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2103 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485390);