#2074071

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
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

1496 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=fhi,d^2=fghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1981267);