#6357192

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357189
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	#60646	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2170 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)(11,19)(12,17)(13,25)(14,16)(15,21)(18,26)(20,32)(22,31)(23,29)(24,37)(27,35)(28,34)(30,36)(33,39)(38,40);(9,11,13,21,14,22,26,35)(10,15,18,19,16,27,25,31)(12,20,24,34,23,33,36,40)(17,28,30,32,29,38,37,39);(1,2,3,5,4,6,7,8)(10,16)(15,27)(17,29)(18,25)(19,31)(28,38)(30,37)(32,39);(9,12)(10,17)(11,20)(13,24)(14,23)(15,28)(16,29)(18,30)(19,32)(21,34)(22,33)(25,37)(26,36)(27,38)(31,39)(35,40);(12,23)(17,29)(20,33)(24,36)(28,38)(30,37)(32,39)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264388);