#8445870

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8353066);