#6447702

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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 ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3072 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=g,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6354898);