#3531668

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

1848 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=gh,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, 3438864);