#4279649

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
automorphism	Aut(G) order 1048576
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×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2352 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=f,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4186845);