#482903

order	512
nilpotency class	4
isoclinism	isoclinic class of #482734
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×C₄⋊C₄	C₂⁴
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#556 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

755 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=be,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fghi,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, 390099);