#614884

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1954 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fgi,c^2=f,d^2=fh,e^2=g,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, 522080);