#6462902

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442590
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60425	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2287 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)(3,8)(4,6)(5,7)(9,10,15,19)(11,20,25,33)(12,26,27,40)(13,17,29,28)(14,30,31,18)(16,24,34,23)(21,22,36,38)(32,39,37,35);(3,4)(6,8)(9,11)(10,16)(12,21)(13,22)(14,24)(15,25)(17,32)(18,33)(19,34)(20,30)(23,31)(26,39)(27,36)(28,37)(29,38)(35,40);(1,3,5,4)(2,6,7,8)(9,12,14,28,15,27,31,17)(10,13,18,26,19,29,30,40)(11,21,24,37,25,36,23,32)(16,22,33,39,34,38,20,35);(1,4,5,3)(2,6,7,8)(9,13)(10,17)(11,22)(12,18)(14,26)(15,29)(16,32)(19,28)(20,36)(21,33)(23,35)(24,39)(25,38)(27,30)(31,40)(34,37);(12,27)(13,29)(17,28)(21,36)(22,38)(26,40)(32,37)(35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6370098);