#4929346

order	512
nilpotency class	2
isoclinism	isoclinic class of #4928752
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×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2367 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,13)(14,15)(17,18)(19,21)(20,22)(23,24)(25,26)(27,31)(28,30)(29,32);(2,5)(6,8)(9,11,14,13)(10,12,15,16)(25,26)(27,29)(28,30)(31,32);(2,5)(6,8)(10,15)(12,16)(17,18,20,22)(19,21,23,24)(25,26,28,30)(27,29,32,31);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,29)(28,30)(31,32);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,12,15,16)(17,19,20,23)(18,21,22,24)(25,27,28,32)(26,29,30,31)

Polycyclic presentation:

2^9|b^2=i,c^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4836542);