#4397046

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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

1608 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)(6,7)(9,10,12,13)(11,14,16,15)(17,18,22,24)(19,20,21,23)(25,26,29,32)(27,33,34,30)(28,36,37,31)(35,38,40,39);(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,19,22,21)(18,20,24,23)(27,34)(30,33)(35,40)(38,39);(2,4)(6,7)(10,13)(14,15)(17,20,22,23)(18,19,24,21)(25,27)(26,30)(28,35)(29,34)(31,38)(32,33)(36,39)(37,40);(1,3,5,8)(2,6,4,7)(9,11)(10,14)(12,16)(13,15)(17,18)(19,20)(21,23)(22,24)(25,26)(27,30)(28,31)(29,32)(33,34)(35,38)(36,37)(39,40);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,21,22,19)(18,23,24,20)(25,28,29,37)(26,31,32,36)(27,35,34,40)(30,38,33,39)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=fhi,d^2=fgi,e^2=fg,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, 4304242);