#5482191

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2377 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)(3,5)(4,6)(7,8)(10,11)(12,16)(17,18)(19,23)(20,22)(21,24)(25,26)(27,31)(28,30)(29,32);(4,8)(6,7)(9,10,13,11)(12,15,16,14)(17,19)(18,21)(20,24)(22,23)(25,27)(26,31)(28,29)(30,32);(2,3)(6,7)(12,16)(14,15)(18,22)(19,24)(26,28)(27,32);(1,3,5,2)(4,7,8,6)(9,11,13,10)(12,15,16,14)(17,19)(18,21)(20,24)(22,23)(25,28,30,26)(27,29,32,31);(1,4)(2,6)(3,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,18)(19,21)(20,22)(23,24)(25,29)(26,27)(28,32)(30,31)

Polycyclic presentation:

2^9|b^2=gi,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5389387);