#6443018

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35328	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2628 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)(9,10,13,19,14,18,27,34)(11,20,22,30,23,35,26,15)(12,17,25,33,24,16,31,32)(21,37,39,28,38,36,40,29);(5,6)(7,8)(9,11,14,23)(10,15,18,30)(12,21,24,38)(13,26,27,22)(16,28,17,29)(19,35,34,20)(25,40,31,39)(32,37,33,36);(1,2,3,4)(6,8)(9,12)(10,16)(11,21)(13,25)(14,24)(15,28)(17,18)(19,32)(20,36)(22,39)(23,38)(26,40)(27,31)(29,30)(33,34)(35,37);(6,8)(9,12)(10,17)(11,21)(13,25)(14,24)(15,29)(16,18)(19,33)(20,37)(22,39)(23,38)(26,40)(27,31)(28,30)(32,34)(35,36);(10,18)(12,24)(15,30)(19,34)(20,35)(21,38)(25,31)(39,40)

Polycyclic presentation:

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

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, 6350214);