#200910

order	512
nilpotency class	3
isoclinism	isoclinic class of #9431
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#386
prime radical	#60688	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1078 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)(3,10,11,14)(4,7,12,13)(5,8)(6,9)(15,16)(17,18)(19,23,29,32)(20,25)(21,27)(22,26)(24,28,31,30);(1,3,5,13,6,11,15,7)(2,4,8,14,9,12,16,10)(17,19)(18,23,26,32)(20,28)(21,30)(22,29)(24,25,31,27);(1,4)(2,7)(3,8)(5,14)(6,12)(9,13)(10,15)(11,16)(17,19)(18,24)(20,28)(21,30)(22,29)(23,25)(26,31)(27,32);(17,20,22,21)(18,25,26,27)(19,28,29,30)(23,31,32,24)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 108106);