#2046004

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

1432 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)(3,8)(4,5)(6,7)(10,12)(13,16)(25,26,30,31)(27,33,35,28)(29,32,37,38)(34,39,40,36);(2,7)(5,8)(9,10,14,12)(11,16,15,13)(18,21)(22,24)(26,31)(28,33)(29,37)(34,40);(1,3,6,4)(2,8,7,5)(9,11)(10,13)(12,16)(14,15)(17,18,20,21)(19,22,23,24)(25,27)(26,28)(29,34)(30,35)(31,33)(32,36)(37,40)(38,39);(1,4,6,3)(2,5,7,8)(9,12,14,10)(11,16,15,13)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,39)(38,40);(1,5)(2,3)(4,7)(6,8)(9,13,14,16)(10,15,12,11)(17,19,20,23)(18,22,21,24)(25,29,30,37)(26,32,31,38)(27,34,35,40)(28,36,33,39)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fgh,d^2=fg,e^2=gh,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, 1953200);