#4032506

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3939702);