#1255343

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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

1738 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=fg,d^2=fgh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1162539);