#5960512

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
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

1721 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=h,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5867708);