#8114440

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) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=hi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8021636);