#5990119

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1642 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)(3,5)(4,7)(6,8)(9,10,12,14)(11,15,16,13)(17,18,22,20)(19,23,24,21)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(11,16)(13,15)(17,19,22,24)(18,23,20,21)(26,29)(31,33)(32,36)(38,39);(2,3)(6,7)(18,20)(19,24)(25,27,30,35)(26,31,29,33)(28,34,37,40)(32,38,36,39);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,22,18)(19,23,24,21)(25,28)(26,32)(27,34)(29,36)(30,37)(31,38)(33,39)(35,40);(1,4,5,8)(2,6,3,7)(9,10)(11,13)(12,14)(15,16)(17,21,22,23)(18,24,20,19)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,39,40,38)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=f,d^2=fg,e^2=g,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, 5897315);