#2795564

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
automorphism	Aut(G) order 1048576
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

1384 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fgi,c^2=fgh,d^2=fh,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef

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, 2702760);