#2052339

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
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	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2234 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fghi,d^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1959535);