#8712479

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

4240 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)(3,7)(4,13)(5,10)(6,8)(9,14)(11,15)(12,16)(17,18,23,22)(19,26,21,24)(20,28,29,25)(27,31,30,32)(33,34)(35,44)(36,46)(37,45)(38,40)(39,41)(42,47)(43,48);(1,3,6,11)(2,7,8,15)(4,12,14,10)(5,13,16,9)(17,19,23,21)(18,24,22,26)(20,27,29,30)(25,31,28,32)(33,35,39,38)(34,40,41,44)(36,43,47,45)(37,42,48,46);(2,8)(3,11)(9,13)(10,12)(20,29)(25,28)(27,30)(31,32)(34,41)(36,47)(40,44)(43,45);(1,3)(2,7)(4,12)(5,9)(6,11)(8,15)(10,14)(13,16)(17,20,23,29)(18,25,22,28)(19,27,21,30)(24,31,26,32)(33,36,39,47)(34,42,41,46)(35,45,38,43)(37,44,48,40);(1,4,6,14)(2,9,8,13)(3,12,11,10)(5,15,16,7)(17,21,23,19)(18,24,22,26)(20,30,29,27)(25,31,28,32)(33,37)(34,43)(35,42)(36,44)(38,46)(39,48)(40,47)(41,45);(1,5)(2,10)(3,13)(4,7)(6,16)(8,12)(9,11)(14,15)(17,22,23,18)(19,26,21,24)(20,25,29,28)(27,31,30,32)(33,38,39,35)(34,40,41,44)(36,45,47,43)(37,42,48,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,d^2=g,e^2=gh,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, 8619675);