#4457608

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

3244 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=h,c^2=h,d^2=i,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,d^c=df

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, 4364804);