#3999733

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×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2277 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=fghi,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, 3906929);