#1687397

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

2486 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=ghi,d^2=ghi,e^2=fgh,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, 1594593);