#8662944

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	#40768	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4119 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:

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

Polycyclic presentation:

2^9|a^2=ghi,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, 8570140);