#6583972

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#35771	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2064 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)(15,19)(16,21)(17,22)(18,23)(20,30)(24,43)(25,34)(26,35)(27,36)(28,39)(29,40)(31,41)(32,42)(33,38)(37,44);(5,7,9,8)(6,10,11,12)(13,15,17,26)(14,19,22,35)(16,28,30,44)(18,32,31,27)(20,37,21,39)(23,42,41,36)(24,33,25,29)(34,40,43,38);(1,2,3,4)(13,16,17,30)(14,20,22,21)(15,24,26,25)(18,33,31,29)(19,34,35,43)(23,40,41,38)(27,44,32,28)(36,39,42,37);(5,8,9,7)(6,10,11,12)(16,29)(18,31)(20,38)(21,40)(23,41)(24,28)(25,44)(27,32)(30,33)(34,37)(36,42)(39,43);(13,16,17,30)(14,21,22,20)(15,25,26,24)(18,33,31,29)(19,34,35,43)(23,38,41,40)(27,28,32,44)(36,39,42,37)

Polycyclic presentation:

2^9|b^2=fi,c^2=g,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491168);