#1964824

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1474 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fghi,c^2=fi,d^2=fi,e^2=fgi,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, 1872020);