#5236500

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
automorphism	Aut(G) order 2097152
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
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2266 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)(3,7)(4,5)(6,8)(9,10,13,11)(12,15,16,14)(17,18,21,20)(19,23,24,22)(25,26)(27,31)(28,30)(29,32);(2,4)(3,8)(9,10,13,11)(12,15,16,14)(17,18)(19,23)(20,21)(22,24)(25,27)(26,31)(28,29)(30,32);(2,4)(6,7)(10,11)(14,15)(18,20)(22,23)(26,28)(27,32);(1,3,5,8)(2,6,4,7)(9,11,13,10)(12,15,16,14)(17,19,21,24)(18,22,20,23)(25,28,30,26)(27,29,32,31);(1,4,5,2)(3,7,8,6)(9,12)(10,14)(11,15)(13,16)(17,20,21,18)(19,23,24,22)(25,29)(26,27)(28,32)(30,31)

Polycyclic presentation:

2^9|a^2=fhi,b^2=gi,d^2=g,e^2=f,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5143696);