#2473239

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

1422 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)(9,10,15,16)(11,12,14,13)(17,18,21,20)(19,22,24,23)(25,26,30,28)(27,31,33,34)(29,32,38,36)(35,39,37,40);(9,11,15,14)(10,13,16,12)(17,19,21,24)(18,22,20,23)(26,28)(27,33)(32,36)(35,37);(9,12,15,13)(10,11,16,14)(17,19)(18,22)(20,23)(21,24)(25,27)(26,31)(28,34)(29,37)(30,33)(32,40)(35,38)(36,39);(1,2)(3,6)(4,5)(7,8)(9,13,15,12)(10,11,16,14)(25,28,30,26)(27,34,33,31)(29,36,38,32)(35,40,37,39);(1,3,4,7)(2,6,5,8)(9,14,15,11)(10,12,16,13)(17,20,21,18)(19,23,24,22)(25,29,30,38)(26,32,28,36)(27,35,33,37)(31,39,34,40)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=g,d^2=f,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, 2380435);