#4416573

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
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

1438 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fi,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=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, 4323769);