#4198651

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1928 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fghi,c^2=fgh,d^2=fh,e^2=f,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, 4105847);