#3966463

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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2738 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)(3,5)(4,7)(6,8)(9,10,13,12)(11,15,16,14)(17,18)(19,23)(20,22)(21,24)(26,28)(27,31);(2,3)(4,8)(10,12)(14,15)(17,18,20,22)(19,23,24,21)(25,26)(27,32)(28,29)(30,31);(4,8)(6,7)(10,12)(14,15)(19,24)(21,23)(26,28)(30,32);(1,3,5,2)(4,6,8,7)(9,11,13,16)(10,14,12,15)(17,18)(19,23)(20,22)(21,24)(25,27)(26,30)(28,32)(29,31);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,14,16,15)(17,19,20,24)(18,21,22,23)(25,28,29,26)(27,30,31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,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, 3873659);