#8581802

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

4016 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=gi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8488998);