#4444417

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1677 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=gi,d^2=hi,e^2=fh,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, 4351613);