#8662888

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	#54839	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4133 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8570084);