#8393750

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

4127 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300946);