#483714

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
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	#3169	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

453 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)(3,7)(4,6)(5,8)(9,12)(10,15)(11,13)(14,16)(17,18,22,27,21,28,36,45,23,25,38,41,37,26,42,44)(19,29,32,34,33,40,47,35,30,43,46,48,31,24,39,20);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19)(18,24)(20,27)(21,33)(22,32)(23,30)(25,40)(26,43)(28,29)(31,37)(34,45)(35,41)(36,47)(38,46)(39,42)(44,48);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,20,22,29,21,34,36,40,23,35,38,43,37,48,42,24)(18,19,27,32,28,33,45,47,25,30,41,46,26,31,44,39);(2,8)(3,10)(6,9)(11,16)(18,25)(19,30)(26,28)(27,41)(31,33)(32,46)(39,47)(44,45)

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390910);