#6576217

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1925 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)(5,6,8,9)(7,10,11,12)(13,14,18,22)(15,24,27,20)(16,21,31,17)(19,23,34,41)(25,43,38,26)(28,42,35,39)(29,32,44,40)(30,37,36,33);(6,9)(10,12)(13,15,18,27)(14,20,22,24)(16,29,31,44)(17,32,21,40)(19,35,34,28)(23,42,41,39)(25,36,38,30)(26,33,43,37);(2,4)(13,16,18,31)(14,21,22,17)(15,25,27,38)(19,36,34,30)(20,26,24,43)(23,33,41,37)(28,44,35,29)(32,42,40,39);(5,7,8,11)(6,10,9,12)(16,30)(17,33)(19,34)(21,37)(23,41)(25,29)(26,40)(28,35)(31,36)(32,43)(38,44)(39,42);(13,17,18,21)(14,16,22,31)(15,26,27,43)(19,37,34,33)(20,38,24,25)(23,36,41,30)(28,32,35,40)(29,42,44,39)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,e^c=ei,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, 6483413);