#3876583

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2138 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)(3,5)(4,7)(6,8)(9,10,14,12)(11,13,16,15)(18,19)(20,24)(27,31)(29,32);(2,3)(4,8)(9,11,14,16)(10,15,12,13)(18,19)(22,23)(25,26,28,30)(27,29,31,32);(4,8)(6,7)(11,16)(13,15)(17,18,21,19)(20,22,24,23);(1,3,5,2)(4,6,8,7)(9,12,14,10)(11,15,16,13)(17,19,21,18)(20,23,24,22)(25,27)(26,29)(28,31)(30,32);(1,4,5,8)(2,6,3,7)(9,13,14,15)(10,11,12,16)(17,20,21,24)(18,22,19,23)

Polycyclic presentation:

2^9|a^2=gh,b^2=fgi,c^2=g,d^2=fg,e^2=fg,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, 3783779);