#481429

order	512
nilpotency class	4
isoclinism	isoclinic class of #481415
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61433	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

711 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=di,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=dh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388625);