#6578125

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2017 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,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, 6485321);