#6270825

order	512
nilpotency class	2
isoclinism	isoclinic class of #6132643
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1647 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fghi,c^2=ghi,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6178021);