#483575

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂³	C₈×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

411 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,22,26,21,28,37,45,23,27,39,44,38,25,42,43)(19,29,32,46,33,41,48,20,30,36,47,35,31,24,40,34);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,23,30)(18,24,27,41)(20,26,34,44)(21,33,38,31)(22,32,39,47)(25,36,28,29)(35,45,46,43)(37,48,42,40);(2,7)(6,15)(8,12)(9,16)(17,20,22,36,21,35,37,24,23,34,39,29,38,46,42,41)(18,19,26,32,28,33,45,48,27,30,44,47,25,31,43,40);(19,30)(20,34)(24,41)(29,36)(31,33)(32,47)(35,46)(40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321257, #72321289, #72321312, …
maximal normal subgroup of	#72321257, #72321289, #72321312, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390771);