#6458259

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437428
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1896 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365455);