#482354

order	512
nilpotency class	4
isoclinism	isoclinic class of #482075
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60673	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

762 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=ghi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cghi,d^a=di,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi

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, 389550);