#1740882

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1828 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)(9,10,12,14)(11,13,15,16)(18,22)(20,23)(25,26)(27,33)(28,32)(29,35)(30,31)(34,40)(36,39)(37,38);(3,8)(5,7)(9,11,12,15)(10,13,14,16)(18,22)(19,24)(26,31)(28,36)(29,33)(34,37);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,22)(19,23,24,20)(25,27)(26,29)(28,37)(30,35)(31,33)(32,40)(34,36)(38,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,22)(21,24)(25,28,30,36)(26,32,31,39)(27,34,35,37)(29,38,33,40);(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,23)(18,19,22,24)(25,29)(26,27)(28,38)(30,33)(31,35)(32,34)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fghi,c^2=h,d^2=f,e^2=h,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, 1648078);