#2029488

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
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

1814 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)(4,5)(6,8)(9,10,13,15)(11,12,16,14)(18,21)(19,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,37)(38,40);(1,2,4,5)(3,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,18,20,21)(19,24,23,22)(26,31)(27,35)(29,33)(36,38);(2,5)(6,7)(10,15)(11,16)(17,18,20,21)(19,22,23,24)(25,27,30,35)(26,32,31,39)(28,38,34,36)(29,37,33,40);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,22)(20,21)(23,24)(25,28,30,34)(26,33,31,29)(27,36,35,38)(32,37,39,40);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,29,30,33)(26,34,31,28)(27,37,35,40)(32,36,39,38)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fghi,c^2=gh,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1936684);