#8720578

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

4228 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)(4,12)(5,9)(6,10)(7,13)(11,15)(14,16)(17,18,23,25)(19,27,21,24)(20,30,31,26)(22,28,32,29)(33,34)(35,39)(36,44)(37,41)(38,42)(40,45)(43,46)(47,48);(1,3)(2,5)(4,11)(6,9)(7,15)(8,10)(12,14)(13,16)(17,19,23,21)(18,24,25,27)(20,28,31,29)(22,26,32,30)(35,42)(38,39)(43,48)(46,47);(4,13)(7,12)(11,16)(14,15)(18,25)(20,31)(24,27)(28,29)(33,35,37,42)(34,38,41,39)(36,43,45,48)(40,46,44,47);(1,4)(2,7)(3,11)(5,15)(6,13)(8,14)(9,16)(10,12)(17,20,23,31)(18,26,25,30)(19,29,21,28)(22,27,32,24)(33,35)(34,39)(36,43)(37,42)(38,41)(40,47)(44,46)(45,48);(1,3)(2,8)(4,11)(5,10)(6,9)(7,14)(12,15)(13,16)(17,21,23,19)(18,27,25,24)(20,28,31,29)(22,26,32,30)(33,36,37,45)(34,40,41,44)(35,43,42,48)(38,46,39,47);(1,5,6,8)(2,9,10,3)(4,14,13,15)(7,11,12,16)(17,22,23,32)(18,28,25,29)(19,26,21,30)(20,24,31,27)(33,34)(35,38)(36,44)(37,41)(39,42)(40,45)(43,47)(46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=i,d^2=hi,e^2=h,f^2=g,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, 8627774);