#8418697

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

4047 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,25)(19,21,24,26)(20,23,31,29)(27,30,28,32)(33,34,39,38)(35,40,43,46)(36,45,47,41)(37,44,48,42);(2,6)(3,9)(8,13)(11,15)(17,19)(18,21)(20,28)(22,24)(23,32)(25,26)(27,31)(29,30)(33,35)(34,40)(36,44)(37,41)(38,46)(39,43)(42,47)(45,48);(4,12)(8,13)(11,15)(14,16)(19,24)(21,26)(27,28)(30,32)(34,38)(35,43)(41,45)(42,44);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,18)(19,26)(20,29)(21,24)(22,25)(23,31)(27,30)(28,32)(33,36)(34,41)(35,44)(37,40)(38,45)(39,47)(42,43)(46,48);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20)(18,23)(19,27)(21,30)(22,31)(24,28)(25,29)(26,32)(33,37,39,48)(34,42,38,44)(35,45,43,41)(36,46,47,40);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,21,22,26)(18,24,25,19)(20,30,31,32)(23,28,29,27)(33,38,39,34)(35,46,43,40)(36,45,47,41)(37,44,48,42)

Polycyclic presentation:

2^9|a^2=i,d^2=g,e^2=h,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, 8325893);