#5249574

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

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

3257 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fg,c^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232586, #72232680, #72232685
maximal normal subgroup of	#72232586, #72232680, #72232685

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5156770);