#4650471

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1865 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=fhi,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=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, 4557667);