#6820045

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1367 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)(4,6)(5,8)(9,10,13,15)(11,14,12,16)(17,18)(19,30)(20,25)(21,29)(22,27)(23,28)(24,33)(26,35)(31,45)(32,39)(34,42)(36,44)(37,43)(38,47)(40,46)(41,48);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,34)(21,32)(22,35)(23,33)(25,40)(26,39)(27,29)(28,30)(31,36)(37,46)(38,44)(41,45)(42,43)(47,48);(9,12,13,11)(10,14,15,16)(17,20,35,48,23,37,21,36)(18,25,26,41,28,43,29,44)(19,31,32,46,33,47,22,34)(24,38,27,42,30,45,39,40);(2,6)(5,7)(17,21,23,35)(18,26,28,29)(19,32,33,22)(20,36,37,48)(24,39,30,27)(25,41,43,44)(31,46,47,34)(38,40,45,42);(3,8)(5,7)(17,22,21,33,23,32,35,19)(18,27,29,24,28,39,26,30)(20,34,36,47,37,46,48,31)(25,42,44,38,43,40,41,45)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6727241);