#9368135

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

4165 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=h,d^2=ghi,e^2=hi,f^2=ghi,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, 9275331);