#8437817

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	#2631	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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=gi,d^2=gh,e^2=h,f^2=h,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, 8345013);