#482809

order	512
nilpotency class	4
isoclinism	isoclinic class of #482728
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#553 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

795 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390005);