#1538220

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
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	#15267	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2711 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)(4,5)(6,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,20)(21,23)(22,24)(26,28)(30,32);(2,5)(6,7)(9,11,13,12)(10,14,15,16)(17,19,21,24)(18,22,23,20)(27,31)(30,32);(3,8)(6,7)(11,12)(14,16)(18,23)(20,22)(25,26,29,28)(27,30,31,32);(1,3,4,8)(2,6,5,7)(9,12,13,11)(10,14,15,16)(17,20,21,22)(18,19,23,24)(25,27)(26,30)(28,32)(29,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,19)(18,22)(20,23)(21,24)(25,28,29,26)(27,32,31,30)

Polycyclic presentation:

2^9|b^2=fi,c^2=fghi,d^2=f,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1445416);