#6019418

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1769 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fgh,c^2=ghi,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, 5926614);