#4396898

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

1486 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=fi,e^2=g,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, 4304094);