#220723

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	#61287	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

809 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=i,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, 127919);