#201686

order	512
nilpotency class	3
isoclinism	isoclinic class of #201678
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1385 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)(3,6,10,14)(4,11,12,7)(5,8)(9,13)(15,16)(17,18)(19,25,27,22)(20,23)(21,24)(26,31,32,29)(28,30);(1,3)(2,6,8,14)(4,9)(5,10)(7,13,11,16)(12,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32);(1,4,5,12)(2,7,8,11)(3,9,10,15)(6,13,14,16)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31);(4,12)(7,11)(9,15)(13,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=gh,d^c=dh,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, 108882);