#6357827

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357820
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60647	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2176 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:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=gh,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^c=di,e^d=eh

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, 6265023);