#482909

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×C₄⋊C₄	C₂⁴
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#552 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

758 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)(17,18,21,26,22,25,32,38)(19,23,29,35,27,33,41,39)(20,24,30,37,31,36,43,46)(28,34,42,45,40,44,48,47);(1,2)(3,7)(4,11)(5,8)(6,12)(9,14)(10,16)(13,15)(17,19)(18,23,25,33)(20,28)(21,29)(22,27)(24,34,36,44)(26,39,38,35)(30,42)(31,40)(32,41)(37,47,46,45)(43,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(19,27)(23,33)(28,40)(29,41)(34,44)(35,39)(42,48)(45,47);(3,9)(7,14)(10,15)(13,16)(17,20,22,31)(18,24,25,36)(19,28,27,40)(21,30,32,43)(23,34,33,44)(26,37,38,46)(29,42,41,48)(35,45,39,47)

Polycyclic presentation:

2^9|a^2=f,b^2=g,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=dhi,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, 390105);