#6442683

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3486	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=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, 6349879);