#8406487

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

3977 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,d^2=gh,e^2=gh,f^2=h,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, 8313683);