#8406486

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3975 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8313682);