#6465559

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3112 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)(9,10,13,17,14,18,26,32)(11,19,21,28,22,31,25,15)(12,16,24,30,23,29,36,39)(20,33,35,38,34,40,37,27);(6,7)(9,11,14,22)(10,15,18,28)(12,20,23,34)(13,25,26,21)(16,27,29,38)(17,31,32,19)(24,37,36,35)(30,40,39,33);(1,2,3,4)(5,6,8,7)(9,12)(10,16)(11,20)(13,24)(14,23)(15,27)(17,30)(18,29)(19,33)(21,35)(22,34)(25,37)(26,36)(28,38)(31,40)(32,39);(5,7,8,6);(12,23)(16,29)(20,34)(24,36)(27,38)(30,39)(33,40)(35,37)

Polycyclic presentation:

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

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, 6372755);