#6524243

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

1473 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)(17,18,22,24)(19,27,31,40)(20,25,32,38)(21,34,28,26)(23,35,36,30)(29,41,43,48)(33,45,42,39)(37,46,47,44);(9,10)(11,14)(12,13)(15,16)(17,19,28,35,22,31,21,30)(18,23,34,40,24,36,26,27)(20,29,42,46,32,43,33,44)(25,37,45,48,38,47,39,41);(2,5)(6,8)(10,13)(11,15)(18,24)(23,36)(25,38)(26,34)(27,40)(37,47)(39,45)(41,48);(9,11,12,15)(10,14,13,16)(17,20,22,32)(18,25,24,38)(19,29,31,43)(21,33,28,42)(23,37,36,47)(26,39,34,45)(27,41,40,48)(30,44,35,46);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(20,32)(25,38)(29,43)(33,42)(37,47)(39,45)(41,48)(44,46)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,d^2=hi,e^2=gi,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, 6431439);