#6345449

order	512
nilpotency class	3
isoclinism	isoclinic class of #61559
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#60940	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2021 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,f^2=h,g^2=i,b^a=bf,e^a=ei,f^a=fh,d^b=di,f^b=fh,d^c=di

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6252645);