#6443160

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3499	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,e^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,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350356);