#4381884

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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

3375 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=f,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4289080);