#6589477

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1513 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)(11,14)(12,13)(15,16)(17,18,21,24)(19,27,30,23)(20,25,34,39)(22,26,35,40)(28,36,31,38)(29,43,45,37)(32,42,33,41)(44,47,46,48);(2,6)(7,8)(17,19,21,30)(18,23,24,27)(20,29,34,45)(22,28,35,31)(25,37,39,43)(26,38,40,36)(32,46,33,44)(41,48,42,47);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(18,24)(19,28)(20,32)(22,35)(23,36)(25,41)(27,38)(29,45)(30,31)(33,34)(39,42)(47,48);(1,3,5,4)(2,7,6,8)(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,29)(21,34)(22,33)(23,37)(24,39)(26,41)(27,43)(28,44)(30,45)(31,46)(32,35)(36,47)(38,48)(40,42);(1,4,5,3)(2,8,6,7)(19,28)(20,33)(22,35)(23,38)(25,41)(26,40)(27,36)(30,31)(32,34)(39,42)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=fg,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, 6496673);