#8581757

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

4127 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)(4,6)(5,8)(7,11)(10,15)(12,13)(14,16)(17,18,22,25)(19,26,20,23)(21,24,29,32)(27,31,28,30)(33,34,39,36)(35,37,40,42)(38,41,46,44)(43,45,47,48);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(19,20)(23,26)(27,28)(30,31)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,48)(46,47);(5,13)(8,12)(10,16)(14,15)(17,19,22,20)(18,23,25,26)(21,27,29,28)(24,30,32,31)(33,34,39,36)(35,42,40,37)(38,41,46,44)(43,48,47,45);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,20,22,19)(18,23,25,26)(21,27,29,28)(24,31,32,30)(33,36,39,34)(35,42,40,37)(38,41,46,44)(43,45,47,48);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,31)(28,30)(29,32)(33,37,39,42)(34,40,36,35)(38,45,46,48)(41,47,44,43);(1,3)(2,9)(4,7)(5,10)(6,11)(8,15)(12,14)(13,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,32)(26,31)(33,38,39,46)(34,41,36,44)(35,43,40,47)(37,45,42,48)

Polycyclic presentation:

2^9|a^2=i,c^2=i,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, 8488953);