#6583694

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1814 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)(5,6,8,9)(7,10,11,12)(13,14,17,22)(15,19,25,35)(16,21,30,20)(18,23,31,41)(24,43,26,34)(27,36,32,42)(28,39,44,37)(29,40,33,38);(6,9)(10,12)(13,15)(14,19)(16,28)(17,25)(18,32)(20,37)(21,39)(22,35)(23,42)(24,29)(26,33)(27,31)(30,44)(34,38)(36,41)(40,43);(1,2,3,4)(13,16,17,30)(14,20,22,21)(15,24,25,26)(18,33,31,29)(19,34,35,43)(23,40,41,38)(27,44,32,28)(36,39,42,37);(5,7,8,11)(6,10,9,12)(15,25)(16,29)(18,31)(19,35)(20,38)(21,40)(23,41)(24,44)(26,28)(30,33)(34,39)(37,43);(13,16,17,30)(14,21,22,20)(15,26,25,24)(18,33,31,29)(19,34,35,43)(23,38,41,40)(27,28,32,44)(36,39,42,37)

Polycyclic presentation:

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

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6490890);