#6576223

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

1987 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=f,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, 6483419);