#4730192

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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

1623 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)(3,8)(4,7)(5,6)(9,10,13,14)(11,15,16,12)(17,18)(19,23)(20,21)(22,24)(26,30)(28,35)(31,36)(34,39);(1,3,6,4)(2,7,5,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,22)(20,23)(21,24)(26,30)(27,33)(32,37)(34,39);(1,3,6,4)(2,8,5,7)(9,12,13,15)(10,11,14,16)(17,19)(18,23)(20,22)(21,24)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40);(1,4,6,3)(2,8,5,7)(9,10)(11,12)(13,14)(15,16)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40);(1,5,6,2)(3,8,4,7)(9,12)(10,11)(13,15)(14,16)(17,20,21,18)(19,23,24,22)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=ghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=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, 4637388);