#9390422

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

3900 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dghi,e^d=egi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9297618);