#4433700

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1546 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fh,c^2=ghi,d^2=g,e^2=h,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, 4340896);