#4416509

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,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, 4323705);