#5034879

order	512
nilpotency class	2
isoclinism	isoclinic class of #5024635
automorphism	Aut(G) order 2097152
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

2629 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:

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

Polycyclic presentation:

2^9|c^2=fi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=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, 4942075);