#3750205

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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

2072 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=i,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3657401);