#512795

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂³×C₁₆
prime radical	C₂×#41	C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

349 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)(9,10,12,17,13,18,23,30,14,19,24,31,25,32,36,39,15,11,16,20,26,21,28,33,27,22,29,34,37,35,38,40);(2,5)(6,8);(9,11,12,20,13,21,23,33,14,22,24,34,25,35,36,40,15,10,16,17,26,18,28,30,27,19,29,31,37,32,38,39);(1,3,4,7)(2,6,5,8)(10,11)(17,20)(18,21)(19,22)(30,33)(31,34)(32,35)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231855, #72231859, #72231860, …
maximal normal subgroup of	#72231855, #72231859, #72231860, …
maximal quotient of	#72322704

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419991);