#4453707

order	512
nilpotency class	2
isoclinism	isoclinic class of #4452694
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

1712 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgi,c^2=fg,d^2=f,e^2=f,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, 4360903);