#6524244

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500450
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

1486 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)(19,28)(20,25)(21,26)(22,36)(23,27)(24,31)(29,38)(30,41)(32,42)(33,39)(34,40)(35,46)(37,45)(43,47)(44,48);(9,10)(11,14)(12,13)(15,16)(17,19,20,29,23,32,22,31)(18,24,36,42,27,38,25,28)(21,30,33,43,34,44,35,45)(26,37,46,48,40,47,39,41);(2,5)(6,8)(10,13)(11,15)(17,20,23,22)(18,25,27,36)(19,29,32,31)(21,33,34,35)(24,28,38,42)(26,39,40,46)(30,43,44,45)(37,41,47,48);(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,30)(20,33)(22,35)(23,34)(24,37)(25,39)(27,40)(28,41)(29,43)(31,45)(32,44)(36,46)(38,47)(42,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(21,34)(26,40)(30,44)(33,35)(37,47)(39,46)(41,48)(43,45)

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,e^b=eh,d^c=dh,e^d=ei

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, 6431440);