#3943038

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

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

2254 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=fi,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, 3850234);