#482795

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	#2631	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

803 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)(17,18,22,27,23,26,35,41)(19,28,30,42,21,24,34,37)(20,25,32,40,33,39,45,48)(29,38,43,47,31,36,44,46);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19,23,21)(18,24,26,28)(20,31,33,29)(22,30,35,34)(25,38,39,36)(27,42,41,37)(32,44,45,43)(40,46,48,47);(1,3)(2,7)(4,11)(5,10)(6,13)(8,15)(9,14)(12,16)(17,20,23,33)(18,25,26,39)(19,29,21,31)(22,32,35,45)(24,36,28,38)(27,40,41,48)(30,43,34,44)(37,46,42,47);(3,10)(7,14)(11,16)(13,15)(17,21,23,19)(18,24,26,28)(20,29,33,31)(22,34,35,30)(25,38,39,36)(27,37,41,42)(32,43,45,44)(40,47,48,46)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=h,f^2=i,b^a=be,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,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, 389991);