#4385975

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1430 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=hi,c^2=f,d^2=g,e^2=h,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, 4293171);