#6819892

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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1372 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)(3,6)(4,8)(5,7)(10,13)(11,15)(17,18,21,25)(19,27,30,23)(20,24,34,40)(22,26,29,36)(28,41,42,35)(31,39,33,37)(32,38,44,47)(43,48,45,46);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(19,28)(20,32)(22,29)(23,35)(24,38)(26,36)(27,41)(30,42)(31,33)(34,44)(37,39)(40,47);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,19,29,42,21,30,22,28)(18,23,36,41,25,27,26,35)(20,33,44,43,34,31,32,45)(24,39,47,46,40,37,38,48);(9,11,12,15)(10,14,13,16)(19,30)(23,27)(28,42)(31,33)(35,41)(37,39)(43,45)(46,48);(17,20,22,32,21,34,29,44)(18,24,26,38,25,40,36,47)(19,31,28,43,30,33,42,45)(23,37,35,46,27,39,41,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=fgh,d^2=f,e^2=fh,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, 6727088);