#238290

order	512
nilpotency class	3
isoclinism	isoclinic class of #238285
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1051 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=hi,f^2=hi,b^a=be,c^a=cf,f^a=fi,g^a=gh,c^b=cg,d^b=di,g^b=gh,e^c=eh,f^c=fhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145486);