#8745738

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

4106 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)(3,4)(5,15)(6,11)(7,9)(8,13)(10,12)(14,16)(17,18)(19,24)(20,23)(21,29)(22,26)(25,31)(27,28)(30,32)(33,34,39,38)(35,42,36,40)(37,41,45,46)(43,48,44,47);(1,3,7,13)(2,8,9,4)(5,6,12,16)(10,11,15,14)(17,19,23,27)(18,24,20,28)(21,26,31,30)(22,25,32,29)(33,35,39,36)(34,40,38,42)(37,43,45,44)(41,47,46,48);(2,9)(4,8)(10,15)(11,14)(18,20)(19,27)(25,29)(26,30)(33,34,39,38)(35,40,36,42)(37,41,45,46)(43,47,44,48);(1,4,7,8)(2,3,9,13)(5,14,12,11)(6,15,16,10)(17,20,23,18)(19,24,27,28)(21,29,31,25)(22,26,32,30)(33,36,39,35)(34,40,38,42)(37,44,45,43)(41,47,46,48);(1,5)(2,10)(3,6)(4,14)(7,12)(8,11)(9,15)(13,16)(17,21)(18,25)(19,26)(20,29)(22,28)(23,31)(24,32)(27,30)(33,37)(34,41)(35,43)(36,44)(38,46)(39,45)(40,47)(42,48);(1,6)(2,11)(3,12)(4,10)(5,13)(7,16)(8,15)(9,14)(17,22,23,32)(18,26,20,30)(19,25,27,29)(21,24,31,28)(33,38,39,34)(35,42,36,40)(37,41,45,46)(43,47,44,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,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, 8652934);