#4757522

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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

1368 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)(11,16)(18,20)(22,24)(25,26,30,28)(27,33,34,31)(29,32,38,36)(35,40,37,39);(9,10)(11,15)(12,16)(13,14)(17,18,21,20)(19,22,23,24)(26,28)(27,34)(32,36)(35,37);(10,14)(12,15)(17,19)(18,22)(20,24)(21,23)(25,27,30,34)(26,31,28,33)(29,37,38,35)(32,40,36,39);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,15,14,12)(25,28,30,26)(27,33,34,31)(29,36,38,32)(35,40,37,39);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,16,14,11)(17,20,21,18)(19,22,23,24)(25,29,30,38)(26,32,28,36)(27,35,34,37)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=g,d^2=h,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4664718);