#8406945

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

4156 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)(3,8)(4,14)(5,9)(6,10)(7,11)(12,16)(13,15)(17,18)(19,23)(20,22)(21,25)(24,27)(26,30)(28,29)(31,32)(34,40)(35,37)(42,47)(43,45);(1,3,7,12)(2,8,11,16)(4,15,9,6)(5,10,14,13)(17,19,22,27)(18,23,20,24)(21,26,29,32)(25,30,28,31)(33,34,39,40)(35,41,37,36)(38,42,46,47)(43,48,45,44);(3,12)(6,15)(8,16)(10,13)(17,18,22,20)(19,24,27,23)(21,28,29,25)(26,30,32,31)(33,35,39,37)(34,41,40,36)(38,45,46,43)(42,44,47,48);(1,4)(2,5)(3,6)(7,9)(8,13)(10,16)(11,14)(12,15)(17,20,22,18)(19,24,27,23)(21,28,29,25)(26,31,32,30)(33,36)(34,35)(37,40)(38,44)(39,41)(42,43)(45,47)(46,48);(1,5)(2,9)(3,13)(4,11)(6,16)(7,14)(8,15)(10,12)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,30)(29,32)(33,37,39,35)(34,41,40,36)(38,45,46,43)(42,48,47,44);(1,6)(2,10)(3,4)(5,16)(7,15)(8,14)(9,12)(11,13)(17,21)(18,25)(19,26)(20,28)(22,29)(23,30)(24,31)(27,32)(33,38,39,46)(34,42,40,47)(35,43,37,45)(36,44,41,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=h,d^2=gh,e^2=g,f^2=g,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, 8314141);