#2060640

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

1494 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=i,d^2=h,e^2=fh,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, 1967836);