#5593473

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2027 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)(3,5,7,8)(9,10,12,13)(11,14,15,16)(17,18)(19,23)(20,22)(21,24)(25,26,31,34)(27,28,29,32)(30,33,39,40)(35,36,37,38);(11,15)(14,16)(17,18)(19,23)(20,22)(21,24)(25,27,31,29)(26,32,34,28)(30,35,39,37)(33,38,40,36);(10,13)(14,16)(17,18,20,22)(19,21,24,23)(25,28,31,32)(26,27,34,29)(30,38,39,36)(33,37,40,35);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,29,31,27)(26,28,34,32)(30,37,39,35)(33,36,40,38);(1,3,4,7)(2,5,6,8)(9,11,12,15)(10,14,13,16)(17,19,20,24)(18,21,22,23)(25,30,31,39)(26,33,34,40)(27,35,29,37)(28,36,32,38)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=f,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5500669);