#8893742

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

4063 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)(3,9)(4,8)(5,6)(7,11)(10,16)(12,15)(13,14)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32)(33,34)(35,40)(36,38)(37,42)(39,41)(43,45)(44,46)(47,48);(1,3)(2,7)(4,12)(5,9)(6,11)(8,16)(10,14)(13,15)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,35,39,44)(34,40,41,46)(36,42,43,48)(37,45,47,38);(2,5)(7,9)(8,13)(15,16)(19,26)(23,30)(27,31)(28,32)(34,41)(35,44)(38,45)(42,48);(1,3)(2,7)(4,12)(5,9)(6,11)(8,16)(10,14)(13,15)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32)(33,36,39,43)(34,38,41,45)(35,42,44,48)(37,46,47,40);(1,4,6,14)(2,8,5,13)(3,10,11,12)(7,15,9,16)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)(33,37)(34,42)(35,38)(36,40)(39,47)(41,48)(43,46)(44,45);(1,5,6,2)(3,9,11,7)(4,13,14,8)(10,16,12,15)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,38)(34,43)(35,37)(36,41)(39,45)(40,48)(42,46)(44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8800938);