#6589581

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1592 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)(11,14)(12,13)(15,16)(17,18,22,27)(19,24,21,26)(20,32,34,25)(23,28,37,45)(29,44,36,39)(30,40,31,41)(33,48,47,42)(35,46,38,43);(2,6)(7,8)(17,19)(18,24)(20,33)(21,22)(23,30)(25,42)(26,27)(28,40)(29,35)(31,37)(32,48)(34,47)(36,38)(39,43)(41,45)(44,46);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,22,34)(18,25,27,32)(19,29,21,36)(23,38,37,35)(24,39,26,44)(28,46,45,43)(30,33,31,47)(40,42,41,48);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,21,22,19)(18,26,27,24)(20,33,34,47)(23,31,37,30)(25,42,32,48)(28,41,45,40)(29,38,36,35)(39,46,44,43);(1,4,5,3)(2,8,6,7)(19,30)(21,31)(23,37)(24,40)(26,41)(28,45)(29,33)(35,38)(36,47)(39,42)(43,46)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6496777);