#4409984

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1755 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fgh,c^2=fgh,d^2=fgh,e^2=fgi,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, 4317180);