#1117561

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
automorphism	?
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

1904 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)(3,7,8,5)(11,14)(12,16)(18,21)(19,23)(25,26,30,31)(27,32,33,28)(29,34,37,38)(35,39,40,36);(3,8)(5,7)(9,10,13,15)(11,12,14,16)(26,31)(27,33)(29,37)(36,39);(1,3)(2,5)(4,8)(6,7)(9,11,13,14)(10,12,15,16)(17,18,20,21)(19,22,23,24)(25,27)(26,32)(28,31)(29,35)(30,33)(34,39)(36,38)(37,40);(17,19)(18,22)(20,23)(21,24)(25,28,30,32)(26,33,31,27)(29,36,37,39)(34,40,38,35);(1,2)(3,5)(4,6)(7,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,22)(20,23)(21,24)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=fgh,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1024757);