#3547123

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 2097152
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

1634 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)(3,6)(4,8)(5,7)(9,10,12,14)(11,15,16,13)(19,23)(22,24)(25,26)(27,30)(28,33)(29,34)(31,36)(32,38)(35,39)(37,40);(1,3,5,8)(2,6,7,4)(9,10,12,14)(11,13,16,15)(18,21)(22,24)(26,27)(31,33)(32,34)(39,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,18,20,21)(19,22,23,24)(25,27,30,26)(28,33,36,31)(29,32,38,34)(35,39,37,40);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19,20,23)(18,22,21,24)(25,28)(26,31)(27,33)(29,37)(30,36)(32,40)(34,39)(35,38);(1,4)(2,3)(5,6)(7,8)(9,11)(10,13)(12,16)(14,15)(25,29,30,38)(26,32,27,34)(28,35,36,37)(31,39,33,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=fg,c^2=fgi,d^2=gi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3454319);