#5073785

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

1499 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fgi,c^2=g,d^2=fg,e^2=gh,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, 4980981);