#5203486

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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

1536 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)(3,8)(4,7)(5,6)(10,13)(11,15)(17,18,21,24)(19,22,20,23)(25,26,29,32)(27,33,34,30)(28,36,37,31)(35,38,40,39);(2,6)(3,7)(9,10)(11,16)(12,13)(14,15)(17,19,21,20)(18,22,24,23)(27,34)(28,37)(30,33)(31,36);(3,7)(4,8)(10,13)(11,15)(17,19,21,20)(18,22,24,23)(25,27,29,34)(26,30,32,33)(28,35,37,40)(31,38,36,39);(1,3,5,7)(2,4,6,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,33)(28,31)(29,32)(30,34)(35,39)(36,37)(38,40);(1,4)(2,7)(3,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20,21,19)(18,23,24,22)(25,28,29,37)(26,31,32,36)(27,35,34,40)(30,38,33,39)

Polycyclic presentation:

2^9|a^2=fi,b^2=fgi,c^2=fi,d^2=fgh,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5110682);