#4389478

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2138 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=h,d^2=gi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,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, 4296674);