#8761973

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8669169);