#6350609

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350525
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2867 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257805);