#484942

order	512
nilpotency class	4
isoclinism	isoclinic class of #484587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#3459	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

765 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18)(19,27)(20,31)(21,25)(22,26)(23,30)(24,32)(28,42)(29,41)(33,48)(34,38)(35,39)(36,46)(37,45)(40,44)(43,47);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,22,29)(18,23,26,37)(20,28,32,44)(21,30,35,45)(24,36,31,47)(25,27,39,41)(33,43,34,46)(38,40,48,42);(17,20,22,32)(18,24)(19,28,29,44)(21,34,35,33)(23,36)(25,38)(26,31)(27,40)(30,46,45,43)(37,47)(39,48)(41,42);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(19,29)(20,32)(23,37)(24,31)(27,41)(30,45)(33,34)(38,48)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392138);