#8479525

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

4191 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)(3,7)(4,13)(5,8)(6,9)(10,16)(11,14)(12,15)(18,24)(19,20)(26,29)(27,28)(33,34)(35,43)(36,42)(37,47)(38,40)(39,41)(44,45)(46,48);(1,3,6,12)(2,7,9,15)(4,11,5,10)(8,16,13,14)(17,18)(19,25)(20,22)(21,29)(23,24)(26,31)(27,30)(28,32)(33,35,40,39)(34,41,38,43)(36,44,47,46)(37,45,42,48);(1,4,6,5)(2,8,9,13)(3,10,12,11)(7,14,15,16)(17,19,23,20)(18,25,24,22)(21,27,31,28)(26,32,29,30)(33,36,40,47)(34,42,38,37)(35,44,39,46)(41,48,43,45);(1,5,6,4)(2,8,9,13)(3,11,12,10)(7,14,15,16)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,37)(34,36)(35,45)(38,47)(39,48)(40,42)(41,44)(43,46);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,38,40,34)(35,41,39,43)(36,37,47,42)(44,48,46,45);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,16)(12,15)(17,22,23,25)(18,20,24,19)(21,30,31,32)(26,28,29,27)(33,39,40,35)(34,41,38,43)(36,46,47,44)(37,45,42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8386721);