#4262417

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

1518 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)(3,7,8,6)(10,13)(11,15)(17,18,21,19)(20,23,24,22)(25,26,30,28)(27,33,35,31)(29,36,37,32)(34,38,40,39);(1,2)(3,7)(4,5)(6,8)(9,10,12,13)(11,14,15,16)(17,19,21,18)(20,23,24,22)(29,37)(32,36)(34,40)(38,39);(2,5)(3,8)(9,10,12,13)(11,16,15,14)(25,27,30,35)(26,31,28,33)(29,34,37,40)(32,38,36,39);(1,2)(3,7)(4,5)(6,8)(9,11,12,15)(10,14,13,16)(17,20,21,24)(18,22,19,23)(25,28,30,26)(27,31,35,33)(29,36,37,32)(34,38,40,39);(1,3,4,8)(2,6,5,7)(9,10)(11,16)(12,13)(14,15)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=fghi,e^2=fgh,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, 4169613);