#5123193

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

1518 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,24)(19,20,21,22)(25,26,29,32)(27,33,34,30)(28,36,37,31)(35,38,40,39);(2,5)(3,7)(11,15)(13,16)(17,19,23,21)(18,20,24,22)(27,34)(28,37)(30,33)(31,36);(2,5)(6,8)(9,10,12,14)(11,13,15,16)(17,20,23,22)(18,19,24,21)(25,27)(26,30)(28,35)(29,34)(31,38)(32,33)(36,39)(37,40);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,21,23,19)(18,20,24,22)(25,26)(27,33)(28,31)(29,32)(30,34)(35,39)(36,37)(38,40);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,13,14,16)(17,22,23,20)(18,19,24,21)(25,28)(26,31)(27,35)(29,37)(30,38)(32,36)(33,39)(34,40)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=fgh,d^2=hi,e^2=fgh,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, 5030389);