#6524420

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,4,5)(3,6,7,8)(17,18)(19,29)(20,25)(21,26)(22,38)(23,28)(24,34)(27,30)(31,40)(32,45)(33,46)(35,42)(36,39)(37,43)(41,47)(44,48);(9,10)(11,14)(12,13)(15,16)(17,19,30,47,23,33,22,34)(18,24,38,46,28,41,27,29)(20,31,37,32,35,48,21,36)(25,39,26,44,42,45,43,40);(2,5)(6,8)(10,13)(11,15)(17,20,23,35)(18,25,28,42)(19,31,33,48)(21,30,37,22)(24,39,41,45)(26,27,43,38)(29,40,46,44)(32,34,36,47);(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,32)(20,30)(22,35)(23,37)(24,40)(25,27)(28,43)(29,45)(31,34)(33,36)(38,42)(39,46)(41,44)(47,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,33)(20,35)(21,37)(24,41)(25,42)(26,43)(29,46)(34,47)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei,e^d=ei

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, 6431616);