#6584019

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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

2054 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)(13,14,17,21)(15,19,25,35)(16,22,30,20)(18,23,31,40)(24,43,26,34)(27,36,32,42)(28,41,44,37)(29,38,33,39);(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,28)(17,25)(18,32)(20,37)(21,35)(22,41)(23,42)(24,29)(26,33)(27,31)(30,44)(34,39)(36,40)(38,43);(1,2,3,4)(13,16,17,30)(14,20,21,22)(15,24,25,26)(18,33,31,29)(19,34,35,43)(23,38,40,39)(27,44,32,28)(36,41,42,37);(5,8,9,7)(6,10,11,12)(14,21)(15,25)(16,29)(18,31)(20,38)(22,39)(24,44)(26,28)(30,33)(34,37)(36,42)(41,43);(13,16,17,30)(14,22,21,20)(15,26,25,24)(18,33,31,29)(19,34,35,43)(23,39,40,38)(27,28,32,44)(36,41,42,37)

Polycyclic presentation:

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

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491215);