#6576455

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	#35757	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2696 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)(7,12)(8,10)(9,11)(13,14,19,24)(15,20,28,37)(16,21,32,41)(17,22,30,39)(18,23,31,40)(25,34,43,44)(26,35,29,38)(27,36,33,42);(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,29)(17,25)(18,33)(19,28)(21,38)(22,34)(23,42)(24,37)(26,32)(27,31)(30,43)(35,41)(36,40)(39,44);(2,4)(13,16)(14,21)(15,25)(17,31)(18,30)(19,32)(20,34)(22,40)(23,39)(24,41)(26,27)(28,43)(29,33)(35,36)(37,44)(38,42);(5,8,9,7)(6,10,11,12)(16,30)(17,32)(18,31)(21,39)(22,41)(23,40)(25,26)(27,33)(29,43)(34,35)(36,42)(38,44);(13,17,19,30)(14,22,24,39)(15,26,28,29)(16,31,32,18)(20,35,37,38)(21,40,41,23)(25,27,43,33)(34,36,44,42)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,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, 6483651);