#2046713

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×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fh,d^2=fh,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, 1953909);