#6465494

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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	C₂×#3035	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2624 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)(11,20)(12,17)(13,18)(14,28)(15,19)(16,23)(21,37)(22,32)(24,40)(25,34)(26,36)(27,35)(29,31)(30,39)(33,38);(6,7)(9,11)(10,16)(12,24)(13,23)(14,29)(15,22)(17,33)(18,32)(19,31)(20,28)(21,26)(25,39)(27,38)(30,36)(34,40)(35,37);(1,2,3,4)(5,6,8,7)(9,12,15,26)(10,17,19,36)(11,21,22,24)(13,27,14,25)(16,30,31,33)(18,35,28,34)(20,37,32,40)(23,39,29,38);(5,7,8,6)(11,22)(16,31)(20,32)(21,24)(23,29)(30,33)(37,40)(38,39);(9,13,15,14)(10,18,19,28)(11,23,22,29)(12,25,26,27)(16,32,31,20)(17,34,36,35)(21,38,24,39)(30,37,33,40)

Polycyclic presentation:

2^9|c^2=gh,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6372690);