#8445868

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	#2813	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3948 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,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, 8353064);