#5396948

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
automorphism	Aut(G) order 1048576
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

1800 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)(9,10,13,11)(12,15,16,14)(17,18,22,20)(19,24,21,23)(25,26)(27,33)(28,32)(29,30)(31,34)(35,39)(36,37)(38,40);(9,10,13,11)(12,15,16,14)(17,18,22,20)(19,23,21,24)(26,30)(27,34)(32,36)(35,40);(10,11)(14,15)(17,19,22,21)(18,23,20,24)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,38)(39,40);(1,2)(3,5)(4,6)(7,8)(9,11,13,10)(12,15,16,14)(17,20,22,18)(19,23,21,24)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,39)(37,40);(1,3,4,7)(2,5,6,8)(9,12)(10,14)(11,15)(13,16)(17,21,22,19)(18,23,20,24)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=fgh,d^2=g,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5304144);