#6457786

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=fgi,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, 6364982);