#3888259

order	512
nilpotency class	2
isoclinism	isoclinic class of #3858029
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

2054 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)(3,4)(5,6)(7,8)(9,10,14,12)(11,13,16,15)(17,18)(19,23)(20,24)(21,22)(25,26,30,29)(27,28,32,31);(2,6)(4,7)(9,11,14,16)(10,15,12,13)(18,21)(20,23)(26,29)(27,32);(2,6)(4,7)(11,16)(13,15)(17,19,22,24)(18,23,21,20)(25,27,30,32)(26,31,29,28);(1,3,5,8)(2,7,6,4)(9,12,14,10)(11,15,16,13)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31);(1,4,5,7)(2,3,6,8)(9,13,14,15)(10,11,12,16)(17,21,22,18)(19,23,24,20)(25,29,30,26)(27,31,32,28)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fgh,d^2=gh,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, 3795455);