#8745764

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

4100 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,d^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8652960);