#8436213

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=g,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, 8343409);