#5486628

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1918 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)(3,6,7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,34)(28,32)(29,33)(30,31)(35,40)(36,37)(38,39);(3,7)(6,8)(9,10)(11,15)(12,13)(14,16)(17,19,20,23)(18,21,22,24)(26,31)(27,33)(32,39)(35,36);(2,5)(6,8)(10,13)(14,15)(18,22)(21,24)(25,27)(26,29)(28,36)(30,33)(31,34)(32,40)(35,38)(37,39);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,28,30,38)(26,32,31,39)(27,35,33,36)(29,37,34,40);(1,3)(2,6)(4,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,21)(20,23)(22,24)(25,29)(26,33)(27,31)(28,37)(30,34)(32,36)(35,39)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5393824);