#1787023

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2396 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fg,d^2=f,e^2=g,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, 1694219);