#6742128

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1547 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fg,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6649324);