#129708

order	512
nilpotency class	3
isoclinism	isoclinic class of #8491
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	#3475	C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

555 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36904);