#6447106

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2120 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=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, 6354302);