#8712703

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

4149 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:

(1,2,7,6)(3,10,5,8)(4,12,13,9)(11,15,14,16)(17,18)(19,24)(20,22)(21,30)(23,25)(26,28)(27,31)(29,32)(33,34)(35,44)(36,46)(37,45)(38,40)(39,41)(42,47)(43,48);(1,3,7,5)(2,8,6,10)(4,11,13,14)(9,15,12,16)(17,19,23,27)(18,24,25,31)(20,21,29,28)(22,30,32,26)(33,35,39,38)(34,40,41,44)(36,43,47,45)(37,42,48,46);(4,13)(9,12)(11,14)(15,16)(18,25)(19,27)(20,29)(26,30)(34,41)(36,47)(40,44)(43,45);(1,4)(2,9)(3,11)(5,14)(6,12)(7,13)(8,15)(10,16)(17,20,23,29)(18,22,25,32)(19,28,27,21)(24,26,31,30)(33,36,39,47)(34,42,41,46)(35,45,38,43)(37,44,48,40);(1,5,7,3)(2,8,6,10)(4,14,13,11)(9,15,12,16)(17,21,23,28)(18,26,25,30)(19,29,27,20)(22,31,32,24)(33,37)(34,43)(35,42)(36,44)(38,46)(39,48)(40,47)(41,45);(1,6,7,2)(3,10,5,8)(4,9,13,12)(11,15,14,16)(17,22)(18,20)(19,30)(21,24)(23,32)(25,29)(26,27)(28,31)(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=h,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, 8619899);