#6584022

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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	#36015	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2054 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491218);