#5119561

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1910 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)(3,7)(4,6)(5,8)(9,10,13,14)(11,12,15,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,30)(28,33)(29,34)(31,35)(32,37)(36,38)(39,40);(1,3)(2,7)(4,5)(6,8)(9,10,13,14)(11,12,15,16)(19,23)(21,24)(26,27)(28,35)(29,37)(38,39);(2,4)(3,8)(10,14)(11,15)(17,18,20,22)(19,21,23,24)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39);(1,4,6,2)(3,5,8,7)(9,11)(10,12)(13,15)(14,16)(17,18)(19,21)(20,22)(23,24)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(1,5)(2,3)(4,8)(6,7)(9,12)(10,11)(13,16)(14,15)(17,19,20,23)(18,21,22,24)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=fgh,d^2=fi,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, 5026757);