#8691838

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

4208 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)(3,8)(4,10)(5,14)(6,12)(7,13)(9,15)(11,16)(18,23)(24,25)(26,29)(31,32)(33,34,38,42)(35,40,36,39)(37,41,46,45)(43,48,44,47);(1,3,7,6)(2,8,13,12)(4,11,15,14)(5,10,16,9)(17,18,22,23)(19,24,20,25)(21,26,30,29)(27,31,28,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46);(1,4)(2,9)(3,14)(5,12)(6,11)(7,15)(8,16)(10,13)(17,19,22,20)(18,24,23,25)(21,27,30,28)(26,31,29,32)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(1,4)(2,10)(3,11)(5,12)(6,14)(7,15)(8,16)(9,13)(17,18)(19,24)(20,25)(21,29)(22,23)(26,30)(27,32)(28,31)(33,34)(35,39)(36,40)(37,45)(38,42)(41,46)(43,48)(44,47);(1,5)(2,11)(3,10)(4,12)(6,9)(7,16)(8,15)(13,14)(17,20,22,19)(18,25,23,24)(21,27,30,28)(26,31,29,32)(33,36,38,35)(34,40,42,39)(37,43,46,44)(41,47,45,48);(1,6,7,3)(2,12,13,8)(4,11,15,14)(5,10,16,9)(17,21)(18,26)(19,27)(20,28)(22,30)(23,29)(24,31)(25,32)(33,37,38,46)(34,41,42,45)(35,43,36,44)(39,47,40,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8599034);