#220550

order	512
nilpotency class	3
isoclinism	isoclinic class of #220548
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#61057	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1144 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127746);