#6588895

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×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1496 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)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,26)(19,25,30,23)(20,32,35,24)(22,27,37,45)(28,39,47,44)(29,41,31,40)(33,42,34,48)(36,46,38,43);(1,3,5,4)(2,6,8,7)(17,19,21,30)(18,23,26,25)(20,33,35,34)(22,29,37,31)(24,42,32,48)(27,40,45,41)(28,38,47,36)(39,46,44,43);(10,13)(14,16)(17,20,21,35)(18,24,26,32)(19,28,30,47)(22,38,37,36)(23,39,25,44)(27,46,45,43)(29,33,31,34)(40,42,41,48);(1,3,5,4)(2,7,8,6)(9,11,12,15)(10,14,13,16)(17,19,21,30)(18,25,26,23)(20,34,35,33)(22,29,37,31)(24,42,32,48)(27,41,45,40)(28,36,47,38)(39,46,44,43);(1,4,5,3)(2,6,8,7)(19,29)(22,37)(23,40)(25,41)(27,45)(28,33)(30,31)(34,47)(36,38)(39,42)(43,46)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fgi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,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, 6496091);