#1716243

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2138 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=fh,d^2=fi,e^2=h,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, 1623439);