#8159904

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

4086 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=hi,e^2=gh,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, 8067100);