#1637739

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

1498 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,c^2=fgi,d^2=fgh,e^2=fghi,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, 1544935);