#6844010

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1303 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)(19,27)(20,25)(21,23)(22,26)(24,30)(28,37)(29,36)(31,41)(32,45)(33,35)(34,40)(38,44)(39,42)(43,47)(46,48);(9,11)(10,13)(12,16)(14,15)(18,23)(19,28)(20,32)(22,29)(24,37)(25,40)(27,39)(30,42)(33,34)(35,45)(38,41)(43,46);(2,6)(7,8)(11,16)(13,15)(17,19,29,42,21,30,22,28)(18,24,36,37,23,27,26,39)(20,31,32,46,33,44,34,43)(25,38,45,47,35,41,40,48);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(18,23)(19,30)(25,35)(26,36)(28,42)(31,44)(40,45)(43,46);(1,4,5,3)(2,8,6,7)(17,20,29,32,21,33,22,34)(18,25,36,45,23,35,26,40)(19,31,42,46,30,44,28,43)(24,38,37,47,27,41,39,48)

Polycyclic presentation:

2^9|a^2=fi,c^2=fh,d^2=gi,e^2=fghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

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, 6751206);