#8745787

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

3964 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)(3,6)(4,9)(5,8)(7,12)(10,14)(11,16)(13,15)(17,18)(19,20)(21,31)(22,27)(23,25)(24,29)(26,28)(30,32)(33,34,38,42)(35,39,45,48)(36,37,41,40)(43,44,47,46);(1,2)(3,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19,23,29)(18,24,25,20)(21,22,28,32)(26,27,31,30)(33,35,38,45)(34,39,42,48)(36,43,41,47)(37,44,40,46);(2,3)(7,9)(8,10)(15,16)(18,25)(20,24)(26,31)(27,30)(33,36,38,41)(34,40,42,37)(35,43,45,47)(39,46,48,44);(1,3,6,2)(4,9,12,7)(5,8,14,10)(11,15,13,16)(17,20,23,24)(18,19,25,29)(21,30,28,27)(22,31,32,26)(33,35)(34,39)(36,43)(37,44)(38,45)(40,46)(41,47)(42,48);(1,4)(2,7)(3,9)(5,13)(6,12)(8,16)(10,15)(11,14)(17,21)(18,26)(19,22)(20,30)(23,28)(24,27)(25,31)(29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,22)(18,27)(19,28)(20,26)(21,29)(23,32)(24,31)(25,30)(33,37,38,40)(34,41,42,36)(35,44,45,46)(39,47,48,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,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, 8652983);