#6524423

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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1467 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)(17,18,23,27)(19,28,31,45)(20,33,21,25)(22,37,29,26)(24,38,41,32)(30,44,35,43)(34,42,36,47)(39,46,40,48);(9,10)(11,14)(12,13)(15,16)(17,19,29,38,23,31,22,32)(18,24,37,45,27,41,26,28)(20,30,36,48,21,35,34,46)(25,39,47,44,33,40,42,43);(2,5)(6,8)(10,13)(11,15)(17,20,23,21)(18,25,27,33)(19,30,31,35)(22,34,29,36)(24,39,41,40)(26,42,37,47)(28,43,45,44)(32,46,38,48);(9,11,12,15)(10,14,13,16)(17,21,23,20)(18,25,27,33)(19,30,31,35)(22,36,29,34)(24,40,41,39)(26,42,37,47)(28,44,45,43)(32,46,38,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,31)(20,21)(24,41)(25,33)(28,45)(32,38)(34,36)(42,47)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=hi,e^2=gi,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, 6431619);