#201765

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	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1134 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

(1,2,7,11,6,10,5,9)(3,4,12,14,13,15,8,16)(17,18)(19,25,27,22)(20,23)(21,24)(26,31,32,29)(28,30);(1,3,5,12,6,13,7,8)(2,4,11,16,10,15,9,14)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32);(1,4)(2,8)(3,11)(5,16)(6,15)(7,14)(9,13)(10,12)(17,19)(18,22)(20,26)(21,27)(23,29)(24,25)(28,32)(30,31);(1,5,6,7)(2,9,10,11)(3,12,13,8)(4,14,15,16)(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=egh,d^2=hi,g^2=h,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, 108961);