#201844

order	512
nilpotency class	3
isoclinism	isoclinic class of #201701
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1304 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 109040);