#6438821

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3140 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,g^c=gi,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, 6346017);