#4416508

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 2097152
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

1880 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)(3,8)(4,7)(5,6)(9,10)(11,14)(12,13)(15,16)(19,23)(21,24)(27,31)(29,32);(1,3,6,4)(2,7,5,8)(10,13)(14,15)(17,18,20,22)(19,21,23,24);(2,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,21)(20,22)(23,24)(25,26,28,30)(27,29,31,32);(1,4,6,3)(2,8,5,7)(25,27)(26,29)(28,31)(30,32);(1,5,6,2)(3,8,4,7)(9,11,12,16)(10,14,13,15)(17,19)(18,21)(20,23)(22,24)

Polycyclic presentation:

2^9|b^2=fh,c^2=gi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,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, 4323704);