#6840632

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1455 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=fgi,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,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, 6747828);