#4383144

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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

1768 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)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(27,32)(28,30);(1,2,5,4)(3,7,8,6)(9,10,13,12)(11,15,16,14)(17,19,23,22)(18,20,24,21)(25,26,29,31)(27,30,32,28);(2,4)(6,7)(10,12)(14,15)(17,20,23,21)(18,19,24,22)(25,27)(26,30)(28,31)(29,32);(1,3)(2,6)(4,7)(5,8)(9,11,13,16)(10,14,12,15)(17,21,23,20)(18,19,24,22)(25,28,29,30)(26,27,31,32);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,15,16,14)(17,22,23,19)(18,20,24,21)

Polycyclic presentation:

2^9|b^2=fi,c^2=fh,d^2=g,e^2=f,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, 4290340);