#8893444

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

4299 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^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, 8800640);