#8463440

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3967 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=h,e^2=gi,f^2=hi,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, 8370636);