#5177420

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

1572 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=fgh,d^2=f,e^2=fhi,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, 5084616);