#1906890

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

1578 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)(3,5,7,8)(10,11)(14,15)(17,18,21,24)(19,20,23,22)(26,30)(27,33)(28,35)(38,40);(9,10,13,11)(12,15,16,14)(17,19,21,23)(18,22,24,20)(26,30)(27,33)(32,37)(34,39);(10,11)(14,15)(17,20,21,22)(18,23,24,19)(25,26)(27,31)(28,32)(29,30)(33,36)(34,38)(35,37)(39,40);(1,2)(3,5)(4,6)(7,8)(9,11,13,10)(12,15,16,14)(17,20)(18,19)(21,22)(23,24)(25,27,29,33)(26,31,30,36)(28,34,35,39)(32,38,37,40);(1,3,4,7)(2,5,6,8)(9,12,13,16)(10,14,11,15)(17,19)(18,20)(21,23)(22,24)(25,28)(26,32)(27,34)(29,35)(30,37)(31,38)(33,39)(36,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=fgh,d^2=h,e^2=fhi,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, 1814086);