#5023226

order	512
nilpotency class	2
isoclinism	isoclinic class of #5020720
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2331 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgi,c^2=fg,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^c=dg,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, 4930422);