#1507009

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2454 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:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=fg,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, 1414205);