#200613

order	512
nilpotency class	3
isoclinism	isoclinic class of #200430
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	#37616	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1165 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)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18,23,25,22,19,21,26)(20,24,28,32,30,27,29,31);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16)(17,19,22,18)(20,27,30,24)(21,26,23,25)(28,32,29,31);(1,2)(3,7)(4,14)(5,8)(6,9)(10,12)(11,16)(13,15)(17,20)(18,24)(19,27)(21,28)(22,30)(23,29)(25,31)(26,32);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,21,22,23)(18,25,19,26)(20,28,30,29)(24,31,27,32)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=hi,e^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=ei,f^b=fi,g^b=gi,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, 107809);