#6832427

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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

1231 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)(9,10,12,13)(11,14,15,16)(17,18)(19,25)(20,33)(21,27)(22,28)(23,29)(24,30)(26,35)(31,40)(32,41)(34,44)(36,43)(37,42)(38,45)(39,46)(47,48);(1,3,4,8)(2,5,6,7)(17,19)(18,25)(20,21)(22,34)(23,32)(24,35)(26,30)(27,33)(28,44)(29,41)(31,39)(36,47)(37,38)(40,46)(42,45)(43,48);(10,13)(14,16)(17,20,21,19,23,35,24,32)(18,26,27,41,29,33,30,25)(22,37,36,47,38,34,39,31)(28,44,43,40,45,42,46,48);(2,6)(5,7)(17,21,23,24)(18,27,29,30)(19,20,32,35)(22,36,38,39)(25,33,41,26)(28,43,45,46)(31,34,47,37)(40,44,48,42);(3,8)(5,7)(9,11,12,15)(10,14,13,16)(17,22,21,36,23,38,24,39)(18,28,27,43,29,45,30,46)(19,31,35,37,32,47,20,34)(25,40,26,42,41,48,33,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=hi,d^2=i,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6739623);