#6585083

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	#25575	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1865 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)(7,11)(8,12)(9,10)(13,14,17,21)(15,23,26,19)(16,20,31,39)(18,22,32,40)(24,35,27,33)(25,36,28,34)(29,37,30,38)(41,44,42,43);(6,10)(11,12)(13,15,17,26)(14,19,21,23)(16,28,31,25)(18,24,32,27)(20,36,39,34)(22,33,40,35)(29,41,30,42)(37,43,38,44);(1,2,3,4)(15,24)(16,29)(18,32)(19,33)(20,37)(22,40)(23,35)(25,28)(26,27)(30,31)(34,36)(38,39);(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,25)(17,31)(18,30)(19,34)(21,39)(22,38)(23,36)(24,41)(26,28)(27,42)(29,32)(33,43)(35,44)(37,40);(5,8,9,7)(6,12,10,11)(15,24)(16,30)(18,32)(19,33)(20,38)(22,40)(23,35)(26,27)(29,31)(37,39)(41,42)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,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, 6492279);