#6457784

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1488 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,e^c=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6364980);