#129710

order	512
nilpotency class	3
isoclinism	isoclinic class of #8412
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1779
prime radical	#3480	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

569 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dg,e^b=ei,d^c=di,e^c=ei,f^c=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36906);