#4227075

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fg,c^2=fghi,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,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, 4134271);