#8406920

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

4189 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)(17,18,22,23)(19,26,20,24)(21,25,29,30)(27,32,28,31)(33,34)(35,39)(36,38)(37,41)(40,43)(42,46)(44,45)(47,48);(1,3,7,12)(2,8,11,16)(4,15,9,6)(5,10,14,13)(18,23)(24,26)(25,30)(31,32)(33,35,38,43)(34,39,36,40)(37,42,45,48)(41,46,44,47);(3,12)(6,15)(8,16)(10,13)(17,19,22,20)(18,24,23,26)(21,28,29,27)(25,32,30,31)(33,34,38,36)(35,40,43,39)(37,44,45,41)(42,46,48,47);(1,4)(2,5)(3,6)(7,9)(8,13)(10,16)(11,14)(12,15)(17,20,22,19)(18,24,23,26)(21,28,29,27)(25,31,30,32)(33,36,38,34)(35,40,43,39)(37,44,45,41)(42,47,48,46);(1,5)(2,9)(3,13)(4,11)(6,16)(7,14)(8,15)(10,12)(17,18)(19,24)(20,26)(21,25)(22,23)(27,31)(28,32)(29,30)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,6)(2,10)(3,4)(5,16)(7,15)(8,14)(9,12)(11,13)(17,21,22,29)(18,25,23,30)(19,27,20,28)(24,31,26,32)(33,37)(34,41)(35,42)(36,44)(38,45)(39,46)(40,47)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=h,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, 8314116);