#9245149

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

4009 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=g,d^2=gh,e^2=gh,f^2=gh,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, 9152345);