#2568624

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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

1556 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)(6,8)(10,14)(11,12)(25,26,30,29)(27,32,35,31)(28,33,38,37)(34,40,36,39);(9,10)(11,16)(12,15)(13,14)(18,21)(22,24)(25,27,30,35)(26,31,29,32)(28,36,38,34)(33,40,37,39);(9,11,13,12)(10,15,14,16)(17,18)(19,22)(20,21)(23,24)(25,27)(26,32)(28,34)(29,31)(30,35)(33,40)(36,38)(37,39);(1,2,4,5)(3,6,7,8)(9,12,13,11)(10,15,14,16)(25,28)(26,33)(27,34)(29,37)(30,38)(31,39)(32,40)(35,36);(1,3,4,7)(2,6,5,8)(9,10)(11,16)(12,15)(13,14)(17,19,20,23)(18,22,21,24)(25,29,30,26)(27,32,35,31)(28,37,38,33)(34,40,36,39)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=g,d^2=i,e^2=hi,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, 2475820);