#220719

order	512
nilpotency class	3
isoclinism	isoclinic class of #220564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61329	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

825 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,d^b=de,e^b=eh,f^c=fh,g^c=ghi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127915);