#8114374

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

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=gi,e^2=i,f^2=g,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, 8021570);