#9368123

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

4161 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=ghi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=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, 9275319);