#6516144

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
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

1670 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fh,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,g^c=gi,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, 6423340);