#4397005

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

1717 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,c^2=hi,d^2=gi,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, 4304201);