#6820653

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	C₂×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1553 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)(4,5)(7,8)(9,10,13,15)(11,14,12,16)(21,34)(26,41)(30,44)(33,36)(35,47)(39,45)(42,48)(43,46);(2,5)(3,7)(9,11,13,12)(10,14,15,16)(17,18)(19,28)(20,25)(21,35)(22,37)(23,27)(24,29)(26,36)(30,45)(31,38)(32,40)(33,41)(34,47)(39,44)(42,43)(46,48);(9,12,13,11)(10,14,15,16)(17,19,22,29,23,31,20,32)(18,24,25,28,27,40,37,38)(21,30,36,43,34,44,33,46)(26,39,35,42,41,45,47,48);(2,5)(6,8)(17,20,23,22)(18,25,27,37)(19,29,31,32)(21,33,34,36)(24,38,40,28)(26,35,41,47)(30,43,44,46)(39,48,45,42);(1,3,4,7)(2,6,5,8)(17,21,20,33,23,34,22,36)(18,26,37,47,27,41,25,35)(19,30,32,46,31,44,29,43)(24,39,38,48,40,45,28,42)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=i,e^2=fhi,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, 6727849);