#8161282

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

4063 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=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, 8068478);