#2052352

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1780 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=ghi,d^2=gh,e^2=f,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, 1959548);