#482771

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	#2543	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#552 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

815 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,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, 389967);