#481874

order	512
nilpotency class	4
isoclinism	isoclinic class of #481819
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	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

877 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,22,27,23,26,35,41)(19,24,30,38,29,39,44,42)(20,31,32,45,21,25,34,40)(28,37,43,48,33,36,46,47);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19)(18,24,26,39)(20,28)(21,33)(22,30)(23,29)(25,36,31,37)(27,42,41,38)(32,43)(34,46)(35,44)(40,48,45,47);(2,7)(6,14)(8,15)(9,16)(17,20,23,21)(18,25,26,31)(19,28,29,33)(22,32,35,34)(24,36,39,37)(27,40,41,45)(30,43,44,46)(38,47,42,48);(17,21,23,20)(18,25,26,31)(19,28,29,33)(22,34,35,32)(24,37,39,36)(27,40,41,45)(30,43,44,46)(38,48,42,47)

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,c^a=ch,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fghi,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, 389070);