#4681599

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2150 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=fg,d^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=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, 4588795);