#1132892

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

1810 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=fi,d^2=gh,e^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, 1040088);