#5999610

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1560 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=fg,c^2=gh,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

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, 5906806);