#615877

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2567 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 523073);