#2148942

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

1572 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=ghi,c^2=g,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=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, 2056138);