#6458370

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1618 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)(17,18,22,27,23,28,36,37)(19,29,20,25,31,40,33,24)(21,26,35,43,34,42,44,47)(30,41,32,39,45,48,46,38);(2,6)(7,8)(17,19,23,31)(18,24,28,25)(20,22,33,36)(21,32,34,46)(26,41,42,48)(27,40,37,29)(30,44,45,35)(38,47,39,43);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20,22,31,23,33,36,19)(18,25,27,40,28,24,37,29)(21,32,35,45,34,46,44,30)(26,39,43,48,42,38,47,41);(1,4,5,3)(2,8,6,7)(17,21,22,35,23,34,36,44)(18,26,27,43,28,42,37,47)(19,30,20,32,31,45,33,46)(24,38,29,41,25,39,40,48);(21,34)(26,42)(30,45)(32,46)(35,44)(38,39)(41,48)(43,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=fg,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365566);