#4443425

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1568 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=fi,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4350621);