#201151

order	512
nilpotency class	3
isoclinism	isoclinic class of #201094
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1066 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)(3,10,11,14)(4,7,12,13)(5,8)(6,9)(15,16)(17,18,23,28,22,27,21,26)(19,24,25,32,29,20,30,31);(1,3,5,13,6,11,15,7)(2,4,8,14,9,12,16,10)(17,19,23,30,22,29,21,25)(18,24,26,32,27,20,28,31);(1,4)(2,7)(3,8)(5,14)(6,12)(9,13)(10,15)(11,16)(17,20,22,24)(18,25,27,30)(19,28,29,26)(21,32,23,31);(17,21,22,23)(18,26,27,28)(19,25,29,30)(20,31,24,32)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=h,e^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=ehi,g^c=ghi

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, 108347);