#220652

order	512
nilpotency class	3
isoclinism	isoclinic class of #220528
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	#61091	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

997 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,3,6,4,5,7,8)(9,10,15,19,16,12,14,26)(11,20,23,17,24,33,30,21)(13,18,28,36,29,25,31,35)(22,37,39,32,40,34,27,38);(9,11,16,24)(10,17,12,21)(13,27,29,39)(14,30,15,23)(18,34,25,37)(19,33,26,20)(22,31,40,28)(32,36,38,35);(2,5)(3,7)(9,12)(10,16)(11,21)(13,25)(14,26)(15,19)(17,24)(18,29)(20,30)(22,38)(23,33)(27,37)(28,36)(31,35)(32,40)(34,39);(9,13,16,29)(10,18,12,25)(11,22,24,40)(14,31,15,28)(17,32,21,38)(19,36,26,35)(20,37,33,34)(23,39,30,27)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127848);