#6841841

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1323 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)(11,15)(12,14)(13,16)(17,18,22,26)(19,29,31,24)(20,33,32,25)(21,36,37,27)(23,28,34,40)(30,39,44,43)(35,41,45,46)(38,48,47,42);(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,26,29)(20,34,32,23)(21,35,37,45)(25,40,33,28)(27,41,36,46)(30,38,44,47)(39,42,43,48);(2,6)(7,8)(10,14)(11,16)(17,20,23,19,22,32,34,31)(18,25,28,24,26,33,40,29)(21,35,38,30,37,45,47,44)(27,41,42,39,36,46,48,43);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(18,26)(21,37)(24,29)(25,33)(28,40)(30,44)(35,45)(38,47);(1,4,5,3)(2,8,6,7)(17,21,34,47,22,37,23,38)(18,27,40,48,26,36,28,42)(19,30,20,35,31,44,32,45)(24,39,25,41,29,43,33,46)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6749037);