#2003438

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

1730 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)(3,6)(4,8)(5,7)(10,13)(14,16)(17,18,21,22)(19,23,24,20)(25,26,29,30)(27,33,34,31)(28,36,37,32)(35,38,40,39);(1,3,5,8)(2,6,7,4)(11,15)(14,16)(19,24)(20,23)(26,30)(27,34)(28,37)(38,39);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,24)(22,23)(25,26)(27,33)(28,32)(29,30)(31,34)(35,39)(36,37)(38,40);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,14,13,16)(17,20)(18,19)(21,23)(22,24)(25,27,29,34)(26,31,30,33)(28,35,37,40)(32,38,36,39);(1,4)(2,3)(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,20)(21,22)(23,24)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,d^2=fgh,e^2=f,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, 1910634);