#9267015

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

3992 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=i,d^2=gi,e^2=h,f^2=g,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, 9174211);