#3943019

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

2214 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)(4,6)(5,8)(9,10,13,14)(11,12,15,16)(17,18,20,21)(19,22,23,24)(25,26)(27,31)(28,30)(29,32);(1,2,4,6)(3,7,8,5)(9,10)(11,12)(13,14)(15,16)(18,21)(19,23)(25,27)(26,29)(28,32)(30,31);(3,8)(5,7)(10,14)(12,16)(17,19)(18,22)(20,23)(21,24)(25,26,28,30)(27,29,32,31);(1,2)(3,7)(4,6)(5,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,29)(28,32)(30,31);(1,3)(2,5)(4,8)(6,7)(9,12)(10,15)(11,14)(13,16)(17,18)(19,24)(20,21)(22,23)(25,26)(27,31)(28,30)(29,32)

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=i,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, 3850215);