#8720744

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

4170 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)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18,22,23)(19,24,28,27)(20,30,31,25)(21,26,32,29)(34,40)(36,45)(41,42)(43,46);(2,3)(7,9)(8,10)(15,16)(17,18,22,23)(19,27,28,24)(20,25,31,30)(21,26,32,29)(33,34)(35,41)(36,38)(37,42)(39,40)(43,47)(44,46)(45,48);(2,3)(7,9)(8,10)(15,16)(18,23)(19,28)(25,30)(26,29)(33,35,39,37)(34,41,40,42)(36,46,45,43)(38,44,48,47);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,19,22,28)(18,24,23,27)(20,29,31,26)(21,25,32,30)(33,36,39,45)(34,38,40,48)(35,43,37,46)(41,47,42,44);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20)(18,25)(19,29)(21,27)(22,31)(23,30)(24,32)(26,28)(33,37,39,35)(34,42,40,41)(36,46,45,43)(38,44,48,47);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21,22,32)(18,26,23,29)(19,30,28,25)(20,24,31,27)(33,38)(34,36)(35,44)(37,47)(39,48)(40,45)(41,46)(42,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=ghi,d^2=gi,e^2=g,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, 8627940);