#8720724

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

4095 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)(3,9)(4,10)(5,8)(6,7)(11,14)(12,13)(15,16)(17,18,22,23)(19,24,28,27)(20,30,31,25)(21,26,32,29)(33,34)(35,41)(36,39)(37,42)(38,44)(40,45)(43,47)(46,48);(2,7)(3,10)(8,16)(11,13)(17,18,22,23)(19,27,28,24)(20,25,31,30)(21,26,32,29)(33,35)(34,38)(36,43)(37,44)(39,46)(40,47)(41,42)(45,48);(2,7)(3,10)(8,16)(11,13)(18,23)(19,28)(25,30)(26,29)(33,36,37,45)(34,39,42,40)(35,43,44,48)(38,46,41,47);(1,3)(2,4)(5,13)(6,10)(7,9)(8,14)(11,15)(12,16)(17,19,22,28)(18,24,23,27)(20,29,31,26)(21,25,32,30)(33,36)(34,40)(35,43)(37,45)(38,47)(39,42)(41,46)(44,48);(1,4)(2,3)(5,14)(6,9)(7,10)(8,13)(11,16)(12,15)(17,20)(18,25)(19,29)(21,27)(22,31)(23,30)(24,32)(26,28)(33,35)(34,41)(36,43)(37,44)(38,42)(39,47)(40,46)(45,48);(1,5,6,15)(2,8,7,16)(3,11,10,13)(4,12,9,14)(17,21,22,32)(18,26,23,29)(19,30,28,25)(20,24,31,27)(33,34)(35,38)(36,39)(37,42)(40,45)(41,44)(43,46)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8627920);