#6577756

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	#36011	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1981 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)(3,4)(5,6)(7,11)(8,12)(9,10)(16,28)(20,36)(25,42)(29,43)(30,31)(34,44)(37,41)(38,40);(6,10)(11,12)(13,14)(15,24)(16,20)(17,21)(18,23)(19,26)(22,33)(25,41)(27,39)(28,36)(29,44)(30,38)(31,40)(32,35)(34,43)(37,42);(2,4)(13,15,18,27)(14,19,23,35)(16,29,28,43)(17,32,33,26)(20,37,36,41)(21,24,22,39)(25,30,42,31)(34,40,44,38);(5,7,9,8)(6,11,10,12)(13,16,18,28)(14,20,23,36)(15,25,27,42)(17,30,33,31)(19,34,35,44)(21,38,22,40)(24,41,39,37)(26,43,32,29);(5,8,9,7)(6,12,10,11)(14,21)(16,30)(17,33)(19,24)(22,23)(25,43)(26,32)(28,31)(29,42)(34,44)(35,39)(38,40)

Polycyclic presentation:

2^9|c^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,h^b=hi,d^c=dh,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, 6484952);