#2580678

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

1510 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=fg,d^2=fhi,e^2=ghi,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, 2487874);