#6844041

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1248 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18,23,28)(19,30,32,24)(20,25,33,41)(21,29,34,26)(22,37,38,27)(31,39,46,45)(35,43,47,40)(36,42,48,44);(9,11)(10,13)(12,16)(14,15)(17,19,23,32)(18,24,28,30)(20,34,33,21)(22,35,38,47)(25,29,41,26)(27,43,37,40)(31,36,46,48)(39,44,45,42);(2,6)(7,8)(11,16)(13,15)(17,20,21,19,23,33,34,32)(18,25,29,30,28,41,26,24)(22,35,36,31,38,47,48,46)(27,40,44,45,37,43,42,39);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(17,21,23,34)(18,26,28,29)(19,20,32,33)(22,36,38,48)(24,25,30,41)(27,42,37,44)(31,35,46,47)(39,40,45,43);(1,4,5,3)(2,8,6,7)(17,22,34,48,23,38,21,36)(18,27,26,42,28,37,29,44)(19,31,20,35,32,46,33,47)(24,39,41,43,30,45,25,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fhi,d^2=gi,e^2=fgh,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

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, 6751237);