#8418163

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

4139 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)(3,8)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(17,18)(19,23)(20,26)(21,27)(22,25)(24,28)(29,32)(30,31)(34,39)(35,42)(41,47)(43,45);(1,3)(2,8)(4,6)(5,15)(7,12)(9,13)(10,14)(11,16)(17,19)(18,25)(20,29)(21,30)(22,24)(23,28)(26,31)(27,32)(33,34,38,39)(35,40,42,36)(37,41,46,47)(43,48,45,44);(3,12)(6,13)(8,16)(10,15)(17,20,24,21)(18,26,28,27)(19,29,22,30)(23,32,25,31)(33,35,38,42)(34,40,39,36)(37,45,46,43)(41,44,47,48);(1,4)(2,5)(3,13)(6,12)(7,9)(8,10)(11,14)(15,16)(17,21,24,20)(18,27,28,26)(19,29,22,30)(23,32,25,31)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,47)(46,48);(1,5)(2,9)(3,10)(4,11)(6,8)(7,14)(12,15)(13,16)(17,22,24,19)(18,25,28,23)(20,30,21,29)(26,31,27,32)(33,34)(35,40)(36,42)(37,41)(38,39)(43,48)(44,45)(46,47);(1,6,7,13)(2,10,11,15)(3,4,12,9)(5,8,14,16)(17,23,24,25)(18,19,28,22)(20,31,21,32)(26,30,27,29)(33,37,38,46)(34,41,39,47)(35,43,42,45)(36,44,40,48)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gh,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8325359);